On the Tamagawa Number Conjecture for Motives Attachedto Modular Forms

Thesis by

Matthew Thomas Gealy

In Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

California Institute of Technology

Pasadena, California

2006

(Defended December 8, 2005)

ii

c© 2006

Matthew Thomas Gealy

All Rights Reserved

iii

To my grandparents

iv

Acknowledgements

I would like to thank my advisor, Matthias Flach, for introducing me to the world of L-functions, and

for guiding me to what would become this thesis. I am equally indebted to Dinakar Ramakrishnan,

for answering many questions on all corners of mathematics.

I owe much to Jennifer Johnson-Leung, David Whitehouse, Meagan Shemenski, and AJ Tolland

for getting me through this process, especially the hardest parts. I would also like to acknowledge

the roles that Benson Farb, Charlie Koppelman, and Prue Ronneberg have played in shaping my

life, mathematically as well as nonmathematically.

Finally thanks to my parents, grandparents, and the rest of my family, for letting this thesis take

me across the country and out of their lives for so long.

v

Abstract

We carry out certain automorphic and l-adic computations, the former extending results of Beilinson

and Scholl and the latter using ideas of Kato and Kings, related to explicit motivic cohomology

classes on modular varieties. Under mild local and global conditions on a modular form, these give

exactly the coordinates of the Deligne and l-adic realizations of said motivic cohomology class in the

eigenspace attached to the modular form (Theorem 4.1.1). Assuming the Kato’s Main Conjecture

and a Leopoldt-type conjecture, we deduce (a weak version of) the Tamagawa Number Conjecture

for the motive attached to a modular form, twisted by a negative integer.

vi

Contents

Acknowledgements iv

Abstract v

1 Introduction 1

1.1 Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 What is a motive? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 The Tamagawa Number Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 A somewhat equivariant Tamagawa Number Conjecture . . . . . . . . . . . . . . . . 6

2 Review of the Motive M(f) 8

2.1 Geometry of M(f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Cohomology of M(f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Construction of Explicit Elements 12

3.1 A Betti cohomology class and l-adic lattices . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Beilinson’s motivic cohomology class . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4 The Main Theorem 15

4.1 Statement of theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Definitions and Preliminary Results 18

5.1 Conventions and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.2 Analytic parameterizations of modular curves . . . . . . . . . . . . . . . . . . . . . . 19

5.3 Deligne cohomology and realizations of Eisenstein symbols . . . . . . . . . . . . . . . 20

5.4 Some representation theory of GL1 and GL2 . . . . . . . . . . . . . . . . . . . . . . 22

5.5 From classical to adelic automorphic forms . . . . . . . . . . . . . . . . . . . . . . . 24

5.6 Whittaker functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6 The Rankin-Selberg Method 27

6.1 Pure tensor decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

vii

7 Lemmas on Eisenstein Series 32

7.1 Bernoulli numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7.2 An explicit kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

8 The Local Computations 40

8.1 Classical calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

8.1.1 The unramified computation . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

8.1.2 Computation at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

8.2 The bad integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

8.2.1 Supercuspidal case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

8.2.2 Principal series or special with two ramified characters . . . . . . . . . . . . . 46

8.2.3 Principal series with one ramified character . . . . . . . . . . . . . . . . . . . 49

8.2.4 Special with conductor p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

9 Overview of Proof of Theorem 4.1.1(ii) 54

9.1 Key propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

10 On Constructions of Kato 56

10.1 Kato elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

10.2 Relation between Galois cohomology classes . . . . . . . . . . . . . . . . . . . . . . . 59

10.3 A lemma of Kings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

10.4 Proof of Lemma 10.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

11 On Kato’s Main Conjecture 67

11.1 Some Iwasawa theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

11.2 Statement of the Main Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

11.3 Descent to twist (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Bibliography 72

1

Chapter 1

Introduction

1.1 Historical remarks

Among all mathematical objects that have been defined and studied over the past centuries, some are

generally considered to be “arithmetic” (with stress on the third syllable, as opposed to the subject

from grade school). There is no precise definition of criteria for differentiating arithmetic questions

from nonarithmetic. However, generally speaking, many arithmetic objects have the property that

they may be studied “one prime at a time,” or, conversely, are deemed arithmetic because their

study gives insight on the distribution or other properties of prime numbers. As an archetypical

example of the former, given a polynomial Q ∈ Z[X1, . . . , Xn] in some number of variables with

integer coefficients, one may try to count the number of solutions to Q modulo p for each prime p.

To many different types of arithmetic objects X, one may attach what is known as an L-function,

L(X, s), which is an analytic function of a complex variable s. The rough recipe for defining L-

functions is, for each prime p, to study the behavior of X “at p,” and use some invariants to define

a local L-factor Lp(X, s). This should be a rational function in p−s. Formally define

L(X, s) =∏p

Lp(X, s).

The definitions are such that the product tends to converge absolutely when Re(s) is large, so

L(X, s) is an actual function. Surprisingly, functions so constructed often have meromorphic, or

even holomorphic, continuation to all of C.

For example, consider the simplest object of study, a single point. To think of a point arithmeti-

cally, at each prime p, we obtain a set of one element. With the benefit of historical hindsight, the

correct local L-factor to assign to a single point is

Lp(∗, s) =1

1− p−s.

2

And so

L(∗, s) =∏p

11− p−s

=∑n≥1

1n2

= ζ(s),

the Riemann zeta function. So general L-functions are some vast generalizations of the Riemann

zeta function. As values of ζ(s) are neither well-understood on the critical line s = 12 + iτ , nor at

s = 2n + 1 a positive odd integer, questions about values of L-functions are already guaranteed to

be nontrivial.

To any number field K, an arithmetic object to be sure, there is defined a Dedekind zeta function

ζK(s) =∑p⊂K

11− Np−s

.

A nineteenth century theorem, the analytic class number formula (ACNF), expresses ζK(0) in terms

of the values

log |u|, u ∈ O∗K ,

along with other more easily understood terms. As the 20th century progressed, many general L-

functions were defined, but as of the early 1970s, it was not clear what an analogues of the ACNF

should be. In the first place, what should be the analogue of the units? And what should play the

role of log | · |, let alone ask how to interpret the other terms in the the expression for ζK(0).

In the early 1970s, Quillen succeeded in defining the higher algebraic K-groups of a regular

scheme. The group of units of a number field is K1(SpecOK), so there was hope that the Ki could

provide appropriate generalizations of units in whatever formula one hoped to find. Furthermore,

(certain summands of) algebraic K-groups define a universal cohomology theory, which carries re-

alization maps into any other reasonable cohomology theory of a scheme. For a number field K,

Spec (OK ⊗ R) is a collection of points, and so the singular (“Betti”) cohomology

H0B(Spec (OK ⊗ R),R)

is a real vector space indexed by places of K. The realization map is exactly log | · |.

In the late 1970s, Bloch studied realizations from the K-theory of an elliptic curve, and conjec-

tured relations to L-values [Bl00]. Inspired by Bloch, Beilinson developed some additional coho-

mological machinery, and was able to formulate a very general conjecture generalizing the ACNF

(up to algebraic factors) to many schemes X over number fields. Furthermore, Beilinson [Bei86]

gave compelling evidence for his conjecture by verifying parts in the case where X is associated to

a modular form.

3

While there was now a concrete conjecture relating special values of L-functions to cohomology

groups of X, the algebraic factors were still unexplained. Finally, in the early 1990s, work by

Bloch-Kato, Fontaine-Perrin-Riou, and others, formulated a precise conjecture predicting a precise

formula (perhaps up to powers of 2, which remain more difficult to pin down). This strengthening

of Beilinson’s work is known as the Tamagawa Number Conjecture (TNC). There has been various

progress on such conjectures for number fields, but, aside from a theorem of Kings on CM elliptic

curves [Ki01], there has been little progress in the intervening years in more geometric situations.

The subject of this thesis is the TNC in the case Beilinson studied, when X is associated to

a modular form. Because the conjectures involve K-groups, which are not known to be finitely

generated in any reasonable generality, the full strength of the statements necessarily can not be

addressed. However, at heart the TNC is a formula describing how certain explicit elements are

positioned with respect to one another in analytic and l-adic contexts, and such statements can

be formulated unconditionally. We complement Beilinson’s calculations to remove all indeterminant

algebraic factors from his formulae, and are also able to relate explicit motivic cohomology classes to

Kato’s Euler system (which is of some interest in its own right). In the end, under some reasonable

hypothesis, we find that the TNC is implied by Kato’s Main Conjecture, in accordance with a

philosophy of Huber and Kings.

1.2 What is a motive?

The word “motive” appears throughout this thesis. In most cases, a “motive” is an object that is

expected to exist as a Chow motive if one assumes enough conjectures. On the other hand, the

cohomlogy of a motive will always be given as a well-defined cohomology group, the L-function

attached to a motive will always be a well-defined L-function, and so on. For the most part, the

role played by the “motive” itself is merely to index various objects that occur, and to underline

what geometry may be attached to each. Thus the reader will hopefully not take umbrage when

idempotents are split or duals of motives taken; all theorems and propositions are made at the

level of cohomology, where each construction makes sense. Much of this paper could also have been

formulated in terms of Voevodsky’s category of motives. We have avoided this approach because the

proofs of theorems ultimately reduce to some very concrete and explicit computation, and we did

not want to introduce a more abstract (and less familiar) framework when the goal is to do exactly

the opposite. Of course, the reader is invited to dream that all motives exist in whatever context

she feels justified.

4

1.3 The Tamagawa Number Conjecture

We will spend the rest of the introduction stating the Tamagawa Number Conjecture in the case

of interest to us. In this range the conjecture can be formulated somewhat more simply than the

most general one; we use a formulation by Kato and which is used by Kings [Ki01]. However, we

adopt some notational differences. For simplicity, we are interested only in the middle dimensional

cohomology of X. We also interchange the role played by a motive and its dual, so that the L-

function that arises is the L-function attached to X, and we allow more leeway in choosing an

integral basis for the Betti cohomology. Comparing to other formulations in print, these changes

should be evident.

Let X be a smooth proper variety of dimension d over a number field K, with ring of integers

OK . Choose a finite set of primes S containing all places of bad reduction, and an odd prime

number l. For each integer r, one may consider the (hypothetical) Chow motive M = hd(X)(−r),

with formal dual M∗(1) = hd(X)(d+ 1 + r). Whether or not M can be physically constructed via

correspondences, there is a well-defined motivic cohomology group

H1(M∗(1)) := Hd+1M (X, d+ 1 + r) := Kd+1+2r(X)(d+1+r)

Q ,

the rth eigenspace for Adams operators acting on the rational vector space Kd+1+2r(X)Q. The

formulation we use is given only at negative integers; that is, assume r > 0. ( So −r is the negative

integer. In fact, one may take r = 0 if dimX > 0, but we will not consider this range for technical

simplicity)

In this range, the Deligne cohomology group may be computed as

H1D(M∗(1)) := Hd+1

D (X,R(d+ 1 + r)) = HdB(X ×Q C, (2πi)d+rR)+,

while the relevant l-adic cohomology group is

H1et(M

∗(1)) := H1(OK [1Sl

], V ∗l (1)),

where

Vl = Hd(X ×K K,Ql(−r))

V ∗l (1) = Hdet(X ×K K,Ql(d+ r + 1))

and the superscript “+” denotes the eigenspace fixed by complex conjugation. There exist regulator

maps

rD : H1(M∗(1))⊗ R→ H1D(M∗(1))

5

rl : H1(M∗(1))⊗Ql → H1et(M

∗(1)).

For each prime p - l of K, the local L-factor is defined as usual as the determinant of Frobenius

acting on the inertial invariants of Vl

Lp(M, s) = detQl

(1− FrpNp−s | V Ip

l ).

For p | l, take

Lp(M, s) = detQl

(1− φ−1p Np−s | Dcris(Vl)).

These rational functions are expected to be independent of the choice of l, hence dependence on l is

suppressed from the notation. Anyway, define

LS(M, s) =∏p/∈S

Lp(M, s).

Let δ be a Q-basis for HdB(X(C), (2πi)d+rQ)+, and let T ∗l (1) be a Zl-lattice in V ∗l (1) so that,

under the standard comparison isomorphism, δ gives a Zl-basis for T ∗l (1)+. Then the Tamagawa

Number Conjecture at l states

Conjecture 1.3.1. [Tamagawa Number Conjecture at l] Let l be an odd prime, X, S, −r < 0 as

above. Assume that LS(M, s) has analytic continuation to all of C, and that, for all p ∈ S,

Lp(M, 0) 6= 0

Then

(a) rD and rl are isomorphisms, and

#H2(OK [1Sl

], T ∗l (1)) <∞

(b) dimH1(M∗(1)) = ords=0LS(M, s)

(c) There exists a Q-basis ξ of H1(M∗(1)) so that, comparing two bases of a real vector space,

rD(ξ) = L∗Sl(M, 0)δ

(d) With the [· : ·] denoting the generalized index of one lattice with respect to another in a Ql-vector

space containing both,

[H1(OK [1Sl

], T ∗l (1)) : rl(ξ)Zl] = #H2(OK [1Sl

], T ∗l (1))

6

This conjecture is easily seen to be independent of all choices. Notice that we have only given

a statement at l; the full TNC would be the compositum of such statements for all primes l, which

would predict the equality of integers up to a sign. The formulation here is preferable, first, because

the question at 2 is significantly more difficult, and, second, because in any case in practice one

verifies prime by prime. Note that one may use the same or similar ξ, and δ for all l, which is in

fact what we will do.

Because there is no hope at the present of proving that the motivic cohomology groups are of

any reasonable size, the following weak Tamagawa Number Conjecture is commonly made

Conjecture 1.3.2 (weak TNC at l). There exists an explicit subspace H1constr(M

∗(1)) ⊂ H1(M∗(1))

so that all statements in Conjecture 1.3.1 hold with H1(M∗(1)) replaced by H1constr(M

∗(1)).

One expects of course that the explicit subspace of constructible elements appearing in this

conjecture is in fact the whole of the motivic cohomology. This conjecture is established by Kings

in [Ki01] when K is a quadratic imaginary field, X = E is an elliptic curve with CM by the full

ring of integers in K, and l 6= 2, 3, under the assumption that the H2 group appearing is already

known to be finite. The argument uses as a key ingredient Rubin’s work on the main conjecture for

imaginary quadratic fields.

1.4 A somewhat equivariant Tamagawa Number Conjecture

We actually wish to prove a slight and obvious variant of the above conjecture, that is better adapted

to what one means by the motive attached to a modular form. Let X, S, r > 0 be as above, and

again write formally M = hd(X)(−r). Let A be a commutative subring of CHd(X ×X)Q, the ring

of endomorphisms of X as a Chow motive. Suppose A is stable under transposition, that λ : A→ F

is a character of A valued in a number field F , and let λ : A→ F be the conjugate character given

by precomposition of λ by transposition. Fix embeddings of F into C and Ql for each l.

Philosophically, λ and λ ought to define idempotents on the Chow motive X, and so one expects

to have direct summands M ⊗A λ of M , M∗(1) ⊗A λ of M∗(1). On a more concrete level, A acts

on all the cohomology groups of M defined in the previous section, and one may consider the λ−

or λ−eigenquotients of these groups, taken with respect to the chosen embeddings according to the

coefficients used. Similarly one may define a partial L-function LS(M ⊗A λ, s). As the regulator

maps are also A-equivariant and descend to these quotients, we may formulate the analogue of

Conj. 1.3.1 as

Conjecture 1.4.1. Notation as above. Let δ⊗Aλ be an F -basis for Hd(X(C), (2πi)d+rQ)+⊗Aλ, and

let T ∗l (1)⊗A λ be a lattice in V ∗l (1)⊗A λ whose +-part is generated by δ⊗A λ. Assume L(M ⊗A λ, s)

7

has analytic continuation to all of C and that, for all p ∈ S,

Lp(M ⊗A λ, 0)

Then

(a) rD ⊗A λ and rl ⊗A λ are isomorphisms, and

#H2(OK [1Sl

], T ∗l (1)⊗A λ) <∞

(b) The dimension of H1(M∗(1)⊗A λ) is equal to the order of vanishing at s = 0 of each component

of LS(M ⊗A λ, s)

(c) There exists an F -basis ξ ⊗A λ of H1(M∗(1) ⊗A λ) so that, comparing two bases of an real or

complex vector space,

rD(ξ)⊗A λ = L∗Sl(M ⊗A λ, 0)(δ ⊗A λ)

(d) With the [· : ·] denoting the generalized index of one lattice with respect to another in a Ql-vector

space containing both,

[H1(OK [1Sl

], T ∗l (1)⊗A λ) : rl(ξ ⊗A λ)OFl] = #H2(OK [

1Sl

], T ∗l (1)⊗A λ)

Finally, we have a conjecture in the form needed, namely

Conjecture 1.4.2. Same as conjecture 1.4.1, but with H1(M∗(1) ⊗A λ) replaced by an explicit

subspace H1constr

As the title of the section suggests, this conjecture is equivariant in the sense that it involves

a motive with an action of its endomorphism algebra. What is usally termed the Equivariant

Tamagawa Number Conjecture, as formulated by Burns and Flach, is a statement about the various

cohomology groups appearing, and how they are related as modules over such an algebra (see

[Fl04]). For a commutative algebra, the formulation here is roughly asking for the λ-component of

this full equivariant conjecture. However, the full conjecture is much stronger than the sum of its

eigencomponents, because the full conjecture also encodes information about congruences between

different components.

In chapter 2, we will review what is meant by the motive attached to modular form, in the

context just given. In chapter 3, we construct the explicit classes ξ and δ that are to be used in

verifying the conjecture. Chapter 4 states the main theorem, giving the results of explicit calculations

that together verify the conjecture un the case at hand. Chapters 5 to 8 carry out the analytic

computations relevant to part (c) of the conjecture, and chapters 9 to 11 address part (d).

8

Chapter 2

Review of the Motive M(f )

2.1 Geometry of M(f)

Let f be a new cuspidal eigenform of weight k + 2 ≥ 2 and level N ≥ 5, f =∑anq

n, and take j

to be a positive integer. It will be convenient to set t = k + 2 + j. We write f =∑anq

n for the

complex conjugate form and F = Q(an) the field of coefficients. Let T be the Q-algebra generated

by Hecke operators Tp, p - N , acting on Sk+2(Γ1(N)), and T′ the Q-algebra of good Hecke operators

acting on Mk+2(Γ1(N)); T′ T. Write λ : T → F for the character attached to f , so λ is the

character attached to f . We consider TNC for the motive M = M(f)(−j), which has (at least

formally) M∗(1) = M(f)(k + j + 2) = M(f)(t).

We begin by reviewing some geometry. Let Y = Y (N) be the modular curve of full level N

structure (We have chosen N ≥ 5 to avoid any possible difficulties with the existence of such fine

moduli schemes). It is geometrically connected over Q(ζN ), but we consider it as a variety /Q. As

usual, X = X(N)/Q is the completion of Y , and Cusps will denote the reduced complement of Y

in X. We will let Isom denote the µ2-torsor whose geometric points consist of a geometric point of

Cusps along with an orientation of the generalized elliptic curve sitting over that cusp.

Let XΓ(N) → Y be the universal elliptic curve, and λ : XkΓ(N) → Y its k-fold self-product over Y .

The (k+1)-dimensional Kuga-Sato varietyKSkΓ(N) is a smooth compactification of XkΓ(N), semistable

over X. KSkΓ(N) was first constructed by Deligne [De71], or it may be viewed as the toroidal

compactification of the mixed Shimura variety XkΓ(N). Similarly, we have λ : XkΓ1(N) → Y1(N) and

the toroidal compactification KSkΓ1(N), semistable over X1(N). The standard idempotent ε [Sch90]

acts fiberwise, hence compatibly on XkΓ(N), KSkΓ(N), XkΓ1(N), and KSkΓ1(N). The reduced complement

of XkΓ(N) in KSkΓ(N) will be denoted KSk,∞Γ(N), and likewise for KSk,∞Γ1(N); these are disjoint unions of

toric varieties.

9

2.2 Cohomology of M(f)

For any regular scheme X, write as usual HiM(X, •) = K2•−i(X)•Q. One has Gysin sequences

· · · → Hk+2M (KSkΓ1(N), t)

i∗−→ Hk+2M (XkΓ1(N), t)

α−→ Hk+1M (KSk,∞Γ1(N), t− 1)→ · · ·

and

· · · → Hk+2M (KSkΓ1(N), t)(ε)

i∗−→ Hk+2M (XkΓ1(N), t)(ε)

α−→ Hk+1M (KSk,∞Γ1(N), t− 1)(ε)→ · · ·

Formally, define

H1(Meq,cusp(t)) = Hk+2M (KSkΓ1(N), t)(ε)

H1(Meq(t)) = Hk+2M (XkΓ1(N), t)(ε)

The usual Hecke correspondences on XkΓ1(N) extend to KSkΓ1(N), and these restrict to correspon-

dences on KSk,∞Γ1(N). Let T denote the endomorphism Q-algebra of the Chow motive KSkΓ1(N)(ε)

generated by the good Tp. (the notation Meq, resp. Meq,cusp, is to suggest a motive that is the sum

of all, resp. all cuspidal, M(f), considered as an equivariant object under the Hecke algebra). Then

λ, λ define characters of T, and the Gysin sequence carries a T-action.

Lemma 2.2.1. The natural map

H1(Meq(t))⊗eT λ→ H1(Meq,cusp(t))⊗eT λ

is an isomorphism.

Proof. We need to show that

HkM(KSk,∞Γ1(N), t− 1)(ε)⊗eT λ = Hk+1

M (KSk,∞Γ1(N), t− 1)(ε)⊗eT λ = 0.

Suppose Z is a component of KSk,∞Γ1(N), that sits over a cusp of Y1(N) defined over Q(ζM ). As in

the proof of [Sch90], Theorem 1.3.3,

Hk+1M (Z, k + 1 + •)(ε) ∼= H•

M(Q(ζM ), j + 1).

Then

HkM(KSk,∞Γ1(N), t− 1)(ε) ∼= H0

M(CuspsΓ1(N), j + 1) = 0

10

and

Hk+1M (KSk,∞Γ1(N), t− 1)(ε) ∼= H1

M(CuspsΓ1(N), j + 1).

By Borel’s theorem, the latter space is a Q-lattice in H0B(CuspsΓ1(N)(C),R(j))+. The action of T

is the action of T′ on weight k + 2 Eisenstein series, where λ does not appear.

We set

H1M(M∗(1)) := H1(Meq,cusp(t))⊗eT λ ∼= H1(Meq(t))⊗eT λ.

The isomorphism will be made more explicit later. The above arguments also hold in l-adic and

Deligne cohomology; set

H1et(M

∗(1)) := H1(Q,Hk+1(KSkΓ1(N),Q,Ql)(t)(ε)⊗eT λ)

∼= H1(Q,Hk+1(XkΓ1(N),Q,Ql)(t)(ε)⊗eT λ)

and

H1D(M∗(1)) := Hk+2

D (KSkΓ1(N)/R,R(t))(ε)⊗eT λ∼= Hk+2

D (XkΓ1(N)/R,R(t))(ε)⊗eT λ.

Let ρD and ρl be the Deligne and l-adic realization maps taking the algebraic K-theory of KSk

or Xk to the respective cohomology theories. We prefer to think of ρD and ρl as being defined by

Chern classes, as due to Soule, Gillet, and Beilinson, although the reader may prefer to think of

motivic cohomology and regulators in terms of Voevodsky’s triangulated categories. Either way,

compatibility with algebraic correspondences induces

ρl,M∗(1) : H1M(M∗(1))→ H1

l (M∗(1))

ρD,M∗(1) : H1M(M∗(1))→ H1

D(M∗(1)).

To summarize what we have just done: the general TNC is formulated for motives hi(X), X

smooth projective, and generalizes in an obvious manner to direct summands thereof. The motive

M(f) is expected to be a direct summand of hk+1(KSkΓ1(N)), cut out using ε and λ. Unfortunately,

while hk+1(KSk) is at least known to exist by [GHM02], attaching an idempotent to λ in the

category of Chow motives is still an open problem in general. However, it is clear that such a

construction would yield exactly the proposed cohomology groups, and we can thus avoid these

technical difficulties by working directly on realizations. For weight 2, the situation is better, but

we will not make use of this. As it will be convenient to work over the open modular curve, we have

reformulated our groups in terms of Xk. (Another approach to working over the open modular curve

11

is to attempt to split the inclusion i∗ via Eisenstein symbols, which will be touched on in §5.3)

To finish this chapter, we give more concrete descriptions of the cohomology groups. Let c ∈

Gal(C/R) be the complex conjugation acting on geometric points; for a real variety X, Hi(X(C))±

denotes the ±-eigenspace for the action of c. According to [Sch88], p. 9 and [Ka04], 11.2.2, setting

H = R1λ(C)∗Z,

H1D(M∗(1)) = Hk+2

D (XkΓ1(N)/R,R(t))(ε)⊗eT λ= Hk+1

B (XkΓ1(N)(C),R(t− 1))(ε)+ ⊗eT λ= H1

B(Y1(N)(C), (SymkHR)(t− 1))+ ⊗eT λ

and

H1l (M

∗(1)) = H1(Q,Hk+1(XkQ,Ql)(t))(ε)Γ1(N) ⊗eT λ= H1(Q,H1(Y1(N)Q,SymkHQl

)(t)⊗eT λ)

=⊕v|l

H1(Q,H1(Y1(N)Q,SymkHQl)(t)⊗eT,v λ).

Here v ranges over the places of F dividing l, and for such a v, we abuse notation and write Fl for

the v-adic completion of F . The tensor product in the last line is over Ql, with λ taking values in Fl,

the Galois representation appearing there is well known to be the two-dimensional Fl-representation

Vl attached to f .

12

Chapter 3

Construction of Explicit Elements

Choose a partition k = k1 + k2, k1 ≥ 0, which will remain fixed throughout. Consider the following

global hypothesis:

(G): k1 + j is even, and L(f, k1 + 1) 6= 0.

According to Rankin (see [JS81], §5.4) and the functional equation, L(f, s) does not vanish on

(0, k+12 ) or (k+3

2 , k + 2). Thus if k 6= 0, 2, there always exists a partition satisfying (G). If k = 2,

(G) holds for j even, and when j is odd, it requires that L(f, 2) 6= 0. For k = 0, (G) can only hold

if j is even and L(f, 1) 6= 0.

3.1 A Betti cohomology class and l-adic lattices

Consider the k + 1-chain

∆ = ∆k1 : (0, i∞)× [0, 1]k → SL2(Z) o Z2k\H × Ck

(τ, t1, . . . , tk) 7→ (τ, τ · t1, . . . , τ · tk1 , tk1+1, . . . , tk1+k2).

Then ∆ represents a class in

Hk+1(KSk(C),KSk,∞(C),Q)(−1)k1 ∼= HBMk+1 (Xk(C),Q)(−1)k1

∼= Hk+1(Xk(C),Q(k + 1))(−1)k1

(for more details on analytic parameterizations of modular varieties, see §5.2). Assuming (G), we

have

(2πi)jε ∆ ∈ Hk+1(Xk(C),R(t− 1))(ε)+.

13

Write δk1 for its image in H1D(M∗(1)).

Set ± = (−1)k2 . By standard comparison theorems, ε ∆ also gives rise to a class (ε ∆)l ∈

H1et(YQ,SymkHQl

), with image δl = δl,k1 ∈ V ±l . We will assume throughout the paper that δl 6= 0,

which we will see later is implied by (G). Let Tl be any Galois-stable lattice in Vl. When δl 6= 0

there is a unique way to replace Tl by a scalar multiple Tl so that δl generates T±l . For each odd l,

such a choice will remain fixed throughout.

Remark: When (G) can not be made to hold, one can more generally choose other modular

symbols δ that have been twisted by finite order Dirichlet characters χ. In the analogue of condition

(G), one requires the parity of j be given in terms of the parity of χ, and that the twisted L-value

L(f, k1 +1, χ) 6= 0; certainly many such χ exist for any f , even uniformly across all forms of a given

weight and level. However, each choice of δ must be accompanied by a compatible choice of ξ in

the following chapter, resulting in another layer of complexity on all computations that follow (the

kernel functions K of Lemma 7.2.1 will depend on χ). Since condition (G) is automatic in weight

6= 2, 4, we have not attempted these computations, but one expects the same techniques to work.

3.2 Beilinson’s motivic cohomology class

As Y is the modular curve with full level N structure, there is a canonical identification of X[N ],

the N -torsion of the universal elliptic over Y , with (Z/N)2. Let e1 and e2 be the canonical basis of

X[N ], corresponding to (1, 0) and (0, 1) in (Z/N)2. It is convenient to interpret e1 and e2 as defining

elements of Q[(Z/N)2]. For l ≥ 0, set

Q(l)[(Z/N)2] = φ : (Z/N)2 → Q : φ(−c,−d) = (−1)lφ(c, d)

and

Q[Isom](l) = H0M(Isom, 0)(l) = f : ( ∗ ∗0 1 )\GL2(Z/N)→ Q : f(−g) = (−1)lf(g).

Then one has the GL2(Z/N)-equivariant horospherical map ([HK99b] Def. 7.5, or elsewhere; note

that the normalization given here differs from the usual one by a factor of N(l!))

%l : Q(l)[(Z/N)2]→ Q[Isom](l)

%l(ψ)(g) =N l + 1(l + 2)

∑(t1,t2)∈(Z/N)2

ψ(g−1t)Bl+2(t2N

).

14

The same name will be given to the composite

% : Q[(Z/N)2]→ Q(l)[(Z/N)2]→ Q[Isom](l),

where the first map is projection to the (−1)l-eigenspace. Let φi = %j+k1(ei). Recall that there

is a GL2-equivariant residue map Resl : H l+1M (Xl, l + 1) → Q[Isom](l), and that it has a canon-

ical right inverse, the Eisenstein symbol map Eisl [Bei86], [HK99b], §2. Then Eisj+ki(φi) ∈

Hk1+j+1M (Xj+ki , j+ki+1). Let π1 : Xk1+j+k2 → Xk1+j be the projection to the first k1+j coordinates,

let π2 : Xk1+j+k2 → Xj+k2 be the projection to the last j+k2 coordinates, and let π : Xk1+j+k2 → Xk

be defined by omitting the “middle” j coordinates. We consider j : Y (N) → Y1(N) to be given on

moduli problems by forgetting the first canonical torsion section.

Define

ξk1 = ε π∗(π∗1Eisk1+j(φ1) ∪ π∗2Eisj+k2(φ2)) ∈ H2+kM (XkΓ(N), t)(ε)

and take

ξki= j∗(ξki

)⊗ λ ∈ Hk+2M (XkΓ1(N), t)(ε).

The subscript ki will often be suppressed; further, we will write ξNki= ξN if we need to make explicit

the level N appearing in the definition. These motivic cohomology classes are a slight generalization

of those appearing in [DS91], §5.7 (where k1 = k, k2 = 0).

15

Chapter 4

The Main Theorem

The abstract modular form f defines an F ⊗ C-valued holomorphic (k + 1)-form ω′0 on XkΓ1(N)(C),

and let ω0 be its pullback to XkΓ(N)(C). Since f has rapid decay at the cusps, integration against

these forms defines functionals (see §5.3 for more details)

〈·, ω0〉 : Hk+2D (XkΓ(N)/R,R(t))(ε)→ F ⊗ C

〈·, ω′0〉 : Hk+2D (XkΓ1(N)/R,R(t))(ε)→ F ⊗ C.

Because ω′0 has eigensystem λ under the action of T and is holomorphic, the pairing 〈·, ω′0〉

descends to an F -linear map

〈·, f〉M∗(1) : H1D(M∗(1))→ C⊗ F

in such a manner that 〈j∗x ⊗ λ, f〉M∗(1) = 〈x, ω0〉. As H1D(M∗(1)) is a rank 1 R ⊗ F -module, its

elements are completely determined by pairing against f .

4.1 Statement of theorem

Note: While every effort has been made to ensure that all stated equalities are exactly true, it is

perhaps safer to regard them as holding up to powers of 2 and −1. As we do not consider the TNC

at the prime 2, such discrepancies anyway do not affect the final result.

Theorem 4.1.1. Let f be a new cuspidal eigenform of weight k ≥ 2, level N ≥ 5 and coefficients

in F . Assume that j > 0 and that (G) holds, and let L(M)(f, s) denote the partial L function with

Euler factors at primes p |M removed. Then

(i) If the local 2-adic representation attached to f is not supercuspidal, then the leading coefficient

16

L(N),∗(f,−j) ∈ C⊗ F is equal to

(−1)k2+ j2+j

22〈ρD,M∗(1)(ξ), f〉M∗(1)

〈δ, f〉M∗(1)

(ii) Take l odd and v | l. Then ρl,M∗(1)(ξ) ∈ H1(Z[ 1Nl ], Vl(t)) ⊂ H1(Q, Vl(t)). Assume

Kato’s Main conjecture for f and the Leopoldt-type conjecture that H2(Z[ 1Nl ], Tl(t)) is finite. Then

H1(Z[ 1Nl ], Tl(t)) is a rank 1 Ol-module. If l - N , the index of Ll(f,−j)−1ρl,M∗(1)(ξ) with respect

to this lattice is #H2(Z[ 1Nl ], Tl(t)). If l | N , the index of ρl,M∗(1)(ξ) in H1(Z[ 1

Nl ], Tl(t)) is equal to

#H2(Z[ 1Nl ], Tl(t))

Considering Conj. 1.4.2 in this case, part (b) is obvious, and part (a) follows from the hypothesis

already made, as in Kings. The calculations of the theorem exactly address parts (c) and (d), so

Corollary 4.1.1. Let f be a new cuspidal eigenform on Γ1(N), of weight k + 2 ≥ 2 and level

N ≥ 5, whose local representation at 2 is not supercuspidal. Let j > 0 and l odd. Assume that

H2(Z[ 1Nl ], Tl(t)) is finite and that (G) holds. Then the weak TNC for M(f)(−j) at l is implied by

Kato’s l-adic main conjecture for f .

We make a few remarks on this result. As mentioned in the introduction, the main result of

[Ki01] is a proof of TNC for the motive attached to a CM elliptic curve, which is the motive M(ψ)

attached to its Hecke character ψ of the CM field. Let fψ be the associated weight 2 CM form.

The constructions of M(fψ) and M(ψ) are not directly comparable, and the statement of Cor. 4.1.1

for fψ is not a simple reworking of Kings’s result for ψ. However, the same L-series arises in both

statements, and the realizations of these motives are closely related, as can be seen in the treatment

of CM forms in [Ka04], §15. In this sense, the above result may be viewed as a generalization of

Kings’s theorem to all modular forms, as well as to Hecke characters of any (geometric) weight. In

his context the necessary Main Conjecture is known by Rubin; when f has CM, the same result

is relevant here. For progress on the main conjecture for more general general modular forms see

[SU02]. The key observation to proving (ii) is in fact the same technical lemma used in Kings’s

paper. Comments on condition (G) were made in §3.1. A treatment of TNC for finite order Hecke

characters of quadratic imaginary fields is given in [Jo05].

The restriction on the level is imposed solely to keep the geometry nice, and does not play a role

in the local calculations. The hypothesis on the 2-adic representation is somewhat more serious from

a technical point of view. For a dihedral supercuspidal representation, essentially the same proof

works, with just a few parameters changed. But for a nondihedral supercuspidal representation, it

does not appear that the newform in the Whittaker model is known explicitly enough to carry out

the necessary computations! Philosophically, though, this hypothesis is superfluous.

17

When k = 0, up to rational multiple, the result in (i) is exactly Beilinson’s theorem in [Bei86].

However, obtaining an exact formula requires making a good choice of the Eisenstein symbols in ξ,

and up to now it was not known in general which explicit choices gave an equality. The new content

here is really that the choices made in §3.1 and §3.2 give a manageable computation; the idea that

bad zeta integrals could be tractable with a good choice of data was motivated by [Wa02] and [Pr04].

Notice that one may verify the formula embedding by embedding, so choose F → C. With respect

to this embedding, f defines a classical automorphic form.

18

Chapter 5

Definitions and PreliminaryResults

The next four chapters comprise the proof of Theorem 4.1.1(i). Broadly speaking, the proof is

nothing more that Rankin’s trick, along with the evaluation of some local zeta integrals. However,

due to the need to keep track of all constants, and due to the lack of other comprehensive discussions

in the literature, we will go into full detail (in particular, we provide the missing details from [DS91]).

This chapter provides all definitions, and the main results that appear in the computation. Chapter 6

is solid computation, reducing (i) to a statement about certain p-adic integrals. Chapter 7 contains

statements and proofs of a number of small computational results remaining from the previous.

Finally, chapter 8 covers the evaluation of the local zeta integrals, with is the only truly new input

here.

The reduction of (i) to evaluating local zeta integrals proceeds in roughly four steps: the expres-

sion 〈ω0, ξ〉 must be interpreted as an integral of concrete differential forms on a complex manifold,

the simplified to an integral of classical modular forms. We translate from classical into adelic

language, obtaining an expression involving adelic automorphic forms. Right-invariance of certain

terms under diagonal matrices will be exploited to somewhat simplify the expression, and then the

Rankin-Selberg method will be applied, obtaining the desired result

5.1 Conventions and notation

First, let us fix notation on subgroups of GL2. B = ( ∗ ∗∗ ) shall denote the upper triangular Borel,

and U = ( 1 ∗1 ) its unipotent radical. Write Z for the center of GL2, T 1 = ( ∗ 1 ) and T 2 = ( 1

∗ ).

Given N > 0, we write

K(N) ⊂ K1(N) ⊂ K0(N) ⊂ GL2(Z)

19

for the groups of matrices congruent to ( 1 00 1 ), ( ∗ ∗0 1 ), and ( ∗ ∗0 ∗ ) modulo N , respectively. Then the

usual arithmetic subgroups Γ(N), Γ1(N), and Γ0(N) are the intersections of the K with SL2(Z). For

? ∈ ∅, 0, 1, write K?(N)p for the image of K?(N) in GL2(Zp); we will often suppress the index p

when clear from context. Finally we let Kp, resp.∏p≤∞Kp, denote the standard maximal compact

subgroups of GL2(Qp), resp. GL2(A). So Kp = K(1)p = GL2(Zp) for p finite, and K∞ = O2(R).

5.2 Analytic parameterizations of modular curves

First, let us fix conventions regarding the analytic parameterization of the complex points of modular

curves. We write Y (N) for the modular curve parameterizing elliptic curves with full level N

structure. Y (N) is geometrically connected over Q(ζN ), but will normally be considered as a scheme

over Q. The group GL2(Z/N) has a natural left action on Y (N), given on the moduli problem by

g · (E, ( e1e2 )) = (E, g · ( e1e2 )).

Define Y1(N) = Y (1, N) = ( ∗ ∗1 )\Y (N). Then Y1(N) is geometrically connected over Q, and has

complex points

Y1(N)(C) ∼= Γ1(N)\H

∼= SL2(Z)\H × (SL2(Z)/Γ1(N)), [(g−1 · τ)] 7→ [τ, g]

∼= SL2(Z)\H ×GL2(Z/N)/( ∗ ∗1 ),

the last map being induced from the canonical projection SL2(Z)→ GL2(Z/N).

To uniformize one component of Y (N)(C), take

ι : H → Y (N)(C), τ 7→ (C/Zτ + Z,τ

N,

1N

).

It is easy to check that if we identify

Y (N)(C) ∼→ SL2(Z)\H ×GL2(Z/N)

g−1 · (ι(τ))← (τ, g),

then the algebraic map Y (N)→ Y1(N) ∼= ( ∗ ∗1 )\Y (N) and the projection

20

GL2(Z/N)→ GL2(Z/N)/( ∗ ∗1 ) induce a commutative diagram

Y (N)(C) ∼−−−−→ SL2(Z)\H ×GL2(Z/N)y yY1(N)(C) ∼−−−−→ SL2(Z)\H × (GL2(Z/N)/( ∗ ∗1 ))

Let ˜Y (N)(C) = ±U(Z)\H × GL2(Z/N), and p : ˜Y (N)(C) → Y (N)(C) be the obvious map. Co-

ordinates on XnΓ(N)(C) are given by (τ, g, z1, . . . , zn), and pulling back along p induces a morphism

p : ˜XnΓ(N)(C)→ XnΓ(N)(C).

5.3 Deligne cohomology and realizations of Eisenstein sym-

bols

Definitions and first properties of Deligne cohomology can be found in [DS91], §2. Let X be a variety

over R, a ≥ 1. Then an element φa of the Deligne cohomology group HaD(X,R(a)) is represented by

smooth R(a − 1)-valued (a − 1)-forms [φa] on X(C), such that 2d[φa] = ωa + (−1)a−1ωa for some

holomorphic a-form ωa. The cup product on these groups is given by

Lemma 5.3.1. ([DS91], p. 180) Let X be a variety over R, and let

φa ∈ HaD(X,R(a)), φb ∈ Hb(X,R(b)).

Let ωa and ωb be holomorphic forms as above. Then

[φa ∪ φb] + (−1)a−1d([φa] ∧ [φb]) = ωa ∧ [φb] + [φa] ∧ ωb.

More specifically, the construction of ξ involves certain motivic cohomology classes Eisj+ki(φi),

where

φi : ( 1 ∗∗ )\GL2(Z/N)→ Q

is defined by

φi(g) = ρk1+j(ei)(g−1 · i∞).

Explicitly, if g = (A BC D ) ∈ GL2(Z/N),

φ1(g) =Nk1+j+1

(k1 + j + 2)Bk1+j+2(

D det g−1

N)

φ2(g) =N j+k2+1

(j + k2 + 2)Bj+k2+2(

−C det g−1

N).

21

Here Bk is the kth Bernoulli polynomial, taking arguments in [0, 1) (mod 1).

As in [Bei86], §2.2 or [HK99b], §7, we have the following analytic formulae for ρD Eisj+ki(φi):

given τ ∈ H and γ = ( a bc d ) ∈ GL2(R), write j(γ, τ) = cτ + d for the usual factor of automorphy.

Letting Sn denote the symmetric group on n letters, and given nonnegative integers r, s, t, define

ηr,ts =∑

σ∈St/SrSt−r

dzσ(0) ∧ · · · ∧ ˆdzs ∧ · · · ∧ dzσ(t)

where there are r holomorphic differentials, t − r antiholomorphic differentials, and St acts on

0, . . . , t − s. Then the class ρDEiski+j(φi) is represented by a smooth ki + j-form on Xki+j(C)

of the form

(τ, g, z1, . . . , zki+j) 7→(2πi)ki+j+1y

(ki + j + 1)

ki+j∑l=0

(l!(ki + j − l)!)ηki+j−l,ki+j0 ·

∑γ∈±U(Z)\SL2(Z)

φi(γg)(cτ + d)l(cτ + d)ki+j−l

|cτ + d|2(ki+j+1)

+ (dτ, dτ terms).

Let Eiski+j(φi) be the weight ki + j + 2 holomorphic Eisenstein series defined by

(τ, g, z1, . . . , zki+j) 7→ (2πi)ki+j(k1 + j)!∑

γ∈±U(Z)\SL2(Z)

φi(γg)j(γ, τ)ki+j+2

dq

q∧ dz1 ∧ · · · ∧ dzki+j .

Then Eiski+j is the real or imaginary part of d[ρDEiski+j ].

We also explain the pairings appearing in Theorem 4.1.1. Recall that for any real variety X, the

Deligne cohomology groups fit into exact sequences

· · · → HiB(X(C),R(j − 1))+ → Hi+1

D (XR,R(j))→ FiljHi+1DR (XR)→ · · ·

As F k+j+2Hk+1(Xk(C),C) = 0 and F k+j+2Hk+2(Xk(C),C) = 0, the standard exact sequences show

that (functorially for π∗, even)

Hk+2D (Xk/R,R(t)) ∼= Hk+1

B (Xk(C),R(t− 1))+.

There are compatible duality pairings

Hk+1(XkΓ(N)(C),C)×Hk+1c (XkΓ(N)(C),C) ∪−−−−→ H2k+2

c (XkΓ(N)(C),C)

β×Id

y yTr = 1(2πi)k+1

RXk

Γ(N)(C)

HBMk+1(X

kΓ(N)(C),C)×Hk+1

c (XkΓ(N)(C),C)R

−−−−→ C

22

Let f be a cusp form of weight k + 2. The form

ω0 = f(τ, g)dq

q∧ dz1 ∧ · · · ∧ dzk

is holomorphic on KSkΓ(N)(C) of top holomorphic degree, and thus represents classes

ω0 ∈ Hk+1(KSkΓ(N)(C),C), ω0,c ∈ Hk+1c (XkΓ(N)(C),C).

On the other hand, the inclusion

i∗ : Hk+1(KSkΓ(N)(C),C)→ Hk+1(XkΓ(N)(C),C)

has a splitting pr with kernel F k+1 ∩ F k+1, where F k+1 = H0(KSkΓ(N)(C),Ωk+1〈KSk,∞〉)(ε). As

ω0 is holomorphic of top degree, 〈η, ω0,c〉 = 0 for all η ∈ ker(pr), and so

〈∆, ω0,c〉 = 〈i∗pr ∆, ω0,c〉 = 〈pr ∆, ω0〉.

I.e., applying pr eliminates any concerns about convergence of the integrals. For the pairing

〈ρD(ξ), ω0,c〉, we regard both ρD(ξ) and ω0,c as smooth (k + 1)-forms, and the cup product is a

wedge product. As ω0,c has exponential decay and ρD(ξ) only moderate growth, again, there is no

difficulty in convergence of the integral.

5.4 Some representation theory of GL1 and GL2

For readers who are not so familiar with the theory of automorphic forms, we will briefly review

some representation theory of GL1 and GL2. A reference is [Bu98].

First, given any finite order Dirichlet character χ, we may associate a finite order Hecke character

χ : Q∗\A∗ → C∗, and vice versa; we will not distinguish notationally. For any Hecke character

χ : Q∗\A∗ → C∗, write χ =∏p χp. To each χp one can attach a local L-factor. If χp is ramified,

Lp(χp, s) = 1, and if χp is unramified, Lp(χp, s) = (1 − χp(p)p−s)−1. If χ∞(−1) = (−1)m, m =

0, 1, then L∞(χ∞, s) = ΓR(s + m) = π−s+m

2 Γ( s+m2 ). One has the global L-function L(χ, s) =∏p<∞ Lp(χp, s) and the completed global L-function Λ(χ, s) =

∏p≤∞ Lp(χp, s). These have analytic

continuation to all of C (with a pole as s+ 1 if χ = | · |s), and one has the functional equation

Λ(χ, s) = ε(s, χ)Λ(χ−1, 1− s).

23

Writing

G(s) =Γ(s)(2π)s

, G∗(n) =Γ∗(n)(2π)n

,

we may rewrite the functional equation as

L(χ, s) =1

2G(n) cos(π( s−m2 ))ε(s, χ)L(χ−1, 1− s).

Let ψ =∏p ψp : A/Q→ C be the standard unramified additive character

ψ∞(x∞) = e2πix∞ , ψp(xp) = e−2πixp .

Then ε(s, χ) is a product of local ε-factors

ε(s, χ) =∏p≤∞

εp(s, χp, ψp).

If p is finite and χp is unramified, εp(s, χp, ψp) = 1, while if χp is ramified of conductor n > 0,

εp(s, χp, ψp) =∫

p−nZ∗p

|x|−sχ−1p (x)ψp(x)dx.

Meanwhile, for χ∞(−1) = (−1)l, (see [JL70], above Theorem 15)

ε∞(s, χ∞, ψ∞) = (i)l2

=

1 l even

i l odd

Now let f be a classical cuspidal modular form. Via the standard method, which will be reviewed

more thoroughly in the next section, f may be viewed as a smooth vector in C∞(GL2(Q)\GL2(A)).

Let $ be the automorphic representation spanned by the right translates of f . If f is a new

eigenform, it is well known that $ is an irreducible automorphic representation, and that $ = ⊗′$p

is a restricted tensor product of irreducible pre-unitary local representations.

To each local representation $p, one can attach a local L-factor Lp($p, s) (the exact definition

need not concern us now). Given an irreducible automorphic representation $ of GL2(Q) (the

only field we consider here), one again has a global L-function L($, s) =∏p<∞ Lp($p, s) and a

completed global L-function Λ($, s) =∏p≤∞ Lp($p, s). These converge for s >> 0 and have

analytic continuation, with a functional equation

Λ($, s) = ε(s,$)Λ($, 1− s) =

(∏p

ε(s,$p, ψp)

)Λ($, 1− s).

24

Here $ is the dual representation to $; if $ is generated by f , $ is generated by f . If f is a new

eigen cusp form of weight l, L(f, s + l−12 ) = L($, s), as we normalize automorphic L-functions to

have center of symmetry 12 , while motivic L-functions are normalized according to the weights of

the Chow motives from which they arise. Again, L∞(f, s) = ΓC(s) = 2Γ(s)(2π)s , so that

L(f, s) =G(l − s)G(s)

ε(s− l − 12

)L(f , l − s).

Also,

ε∞(s,$∞, ψ∞) = il.

We will not give precise definitions of the finite epsilon factors at the moment.

We shall also need to fix conventions for the Haar measures on GL2(R) and GL2(A). As in

[Bo97], §2.9, consider K∞ ⊂ SL2(R), and let dk be the Haar measure on K∞ with total volume

1. Then SL2(R)/K∞ ∼= H, and a Haar measure on SL2(R) is y−2(dx ∧ dy)dk. As GL2(R) is a

topological direct product Z0(R)SL2(R), it has Haar measure dgR,Haar = y−2dz(dx ∧ dy)dk. We

equip each GL2(Qp) with the Haar measure dgp,Haar, for which GL2(Zp) has volume 1, and then

dg = dgR,Haar ⊗⊗dgp,Haar is our Haar measure on GL2(A).

For parameters z, z, x, y on C, dz ∧ dz = −2idx ∧ dy, and with q = e2πiz,

dq

q∧ dqq

= 4π2dz ∧ dz = −8π2idx ∧ dy.

5.5 From classical to adelic automorphic forms

We can mostly follow [Bu98], although the notation is slightly different because we work on the full

Y (N). Let h be a modular form of weight l for Y (N). Then h : H×GL2(Z/N)→ C is a function

satisfying, for any γ ∈ SL2(Z),

h(γ · τ, γg) = j(γ, τ)lh(τ, g).

Then yl2h(τ, g) is a Maass form, in the sense of [Bu98], §3.2. Define

Fh : SL2(Z)\GL+2 (R)×GL2(Z/N)→ C

by (here g∞ = ( a bc d ))

Fh(g∞, g) = h(g∞ · i, g)Im(g∞ · i)l2

(−ci+ d

|ci+ d|

)l:= h(g∞ · i, g)Im(g∞ · i)

l2ϑ(g∞)l.

25

It is easy to check from this formula that F is indeed invariant under left (diagonal) multiplication by

SL2(Z). Finally, given any g ∈ GL2(A), one may write g = γg∞κ with γ ∈ GL2(Q), g∞ ∈ GL+2 (R),

and κ ∈ GL2(Z). Let κ be the reduction of κ in GL2(Z/N). Then the adelic modular form attached

to h is ϕ(g) = ϕh(g) : GL2(Q)\GL2(A)→ C,

ϕ(g) = Fh(g∞, κ).

If one chooses a different decomposition g = γ′g′∞κ′, then γ′γ−1 ∈ SL2(Z), and so ϕ is well-defined.

As usual, the association f ϕh is equivariant for the respective Hecke algebra actions. Note

that, to be slightly more explicit, suppose we were to write g ∈ GL2(A) as (gR, gf ). Then set

c = ( 1sgn(det gR) ). There is a unique b = ( b1 ∗

b2) ∈ U(Z)\B(Q) with bi > 0 such that

g∞ = cb−1gR ∈ GL+2 (R), and κ = cb−1gf ∈ GL2(Z).

Also, notice that

φ2(κ)Im(g∞ · i)j+k2+2

2 ϑ(g∞)j−k2 =

φ2(b−1gf )(b2b1

) j+k2+22

Im(cgR · i)j+k2+2

2 ϑ(cgR)j−k2

:= φ2(gf )φ2,R(gR).

5.6 Whittaker functions

Each irreducible admissible local representation $p has a conductor np; this is the minimum non-

negative integer such that $p possesses a vector fixed by

K1(pnp) = ( a bc d ) ∈ GL2(Zp) : c, d− 1 ∈ pnpZp.

According to Casselman [Ca73], dim$K1(p

np )p = 1, and for n ≥ np, dim$

K1(pn)

p = n − np + 1. A

global irreducible automorphic representation $ has np = 0 for almost all p, and the conductor of

$ is∏p p

np . If $ is generated by f , the level of f and the conductor of $ coincide [Car86].

We also review some facts about Whittaker functions (at least for finite primes; the situation

at the infinite prime is somewhat more technical, as in [JL70]). Recall that ψp is the standard

unramified character of Qp. The space of local Whittaker functions is

26

Wψp= f : GL2(Qp)→ C : f(( 1 x

1 )g) = ψp(x)f(g).

Since all representations of GL2(Qp) are generic, each $p occurs in Wψpas a representation of

GL2(Qp). By multiplicity 1, each $p occurs exactly once. The unique element

W$p ∈ $K1(pnp )

p ⊂ Wψp, W$p

( 11 ) = 1

is called the newvector in the space of local Whittaker functions. When $p is spherical, that is, for

np = 0, ∫Q∗p

W$p( a 1 )|a|s− 1

2 d∗a = Lp($p, s).

For a global irreducible automorphic representation $, there is a global Whittaker model Wψ =

⊗′Wψp, and $ occurs exactly once in Wψ. There is a newvector W$ = ⊗′W$p

∈ $ ⊂ Wψ. When

$ is generated by f , we will write Wf and Wf,p in place of W$ and W$p ; Wf is (in an appropriate

sense) the Fourier expansion of f . Outside of the case of supercuspidal representations at p = 2,

formulae for newforms in models of the $p are well-known; a concise compilation of results can be

found in [Sc02]. This can be taken as a standard reference for later chapters, where such formulae

will be used heavily.

27

Chapter 6

The Rankin-Selberg Method

Let ω0 be the k + 1-form on Xk representing a cusp form (for now, we need not assume any more).

In local coordinates,

ω0 = f(τ, g)dq

q∧ dz1 ∧ · · · ∧ dzk

and f has rapid decay at the cusps; moreover, ε ω0 = ω0. Then

〈ω0, ε π∗(π∗1ρDEisk1+j(φ1) ∪ π∗2ρDEisj+k2(φ2))〉

= 〈ω0, π∗(π∗1ρDEisk1+j(φ1) ∪ π∗2ρDEisj+k2(φ2))〉

= 〈π∗ω0, π∗1ρDEisk1+j(φ1) ∪ π∗2ρDEisj+k2(φ2)〉

=1

(2πi)j+k+1

∫Xj+k

Γ(N)(C)

π∗ω0 ∧ [π∗1ρD Eisk1+j(φ1) ∪ π∗2ρD Eisj+k2(φ2)]

=1

(2πi)j+k+1

∫Xj+k

Γ(N)(C)

π∗ω0 ∧ (π∗1Eisk1+j(φ1) ∧ [π∗2ρD Eisj+k2(φ2)]

+ [π∗1ρD Eisk1+j(φ1)] ∧ π∗2Eisj+k2(φ2))

=1

(2πi)j+k+1

∫Xj+k

Γ(N)(C)

π∗ω0 ∧ π∗1Eisk1+j(φ1) ∧ [π∗2ρD Eisj+k2(φ2)]

=(2πi)−k1k2!j!N(j + k2 + 1)!

∫Xj+k

Γ(N)(C)

π∗ω0 ∧ π∗1Eisk1+j(φ1) ∧ p∗(yφ2(g)

× dzk1+1 ∧ · · · ∧ dzk1+j ∧ dzk1+j+1 ∧ · · · ∧ dzk+j)

=(2πi)−k1k2!j!N(j + k2 + 1)!

∫˜

Xj+kΓ(N)(C)

p∗π∗ω0 ∧ p∗π∗1Eisk1+j(φ1) ∧ (yφ2(g)

× dzk1+1 ∧ · · · ∧ dzk1+j ∧ dzk1+j+1 ∧ · · · ∧ dzk+j

28

(rearranging the differentials)

= (−1)k2+k+j2+j

22jπji−jk2!j!

N2(j + k2 + 1)!

∫˜

Xj+kΓ(N)(C)

f(τ, g)

(∑γ

φ1(γg)j(γ, τ)k1+j+2

)yφ2(g)

×dqq∧ dqq∧ dz1 ∧ dz1 ∧ · · · ∧ dzk1+j+k2 ∧ dzk1+j+k2

=2j+kik

2+2k+j2+j+1G(k2 + 1)(j)!(2πi)N2G(j + k2 + 2)

∫˜Y (N)(C)

f(τ, g)

(∑γ

φ1(γg)(cτ + d)k1+j+2

)yj+k+1φ2(g)

dq

q∧ dqq.

Set

C1 =2j+k+2πik

2+2k+j2+j−1G(k2 + 1)(j)!G(j + k2 + 2)

.

We may rewrite the integral as

C1

N2

∫±U(Z)\H×GL2(Z/N)

f(τ, g)

(∑γ

φ1(γg)(cτ + d)k1+j+2

)yj+k+1φ2(g)dx ∧ dy

=C1

N2

∫±U(Z)\SL2(R)×GL2(Z)/K+

∞K(N)

f(g∞ · i, κ)Eφ1(g∞ · i, κ)φ2(κ)yj+k+3y−2dx ∧ dy

=C1

N2

∫±U(Z)Z(R)\GL2(R)×GL2(Z)/K∞K(N)

Ff (g∞, κ)FE(g∞, κ)φ2(κ)Im(g∞ · i)j+k2+2

2 ϑ(g∞)j−k2dg∞

=C1

N2

∫B(Q)Z(R)\GL2(A)/K∞K(N)

ϕf (g)ϕE(g)φ2(gf )φ2,R(gR)dgR

= C1[GL2(Z) : K(N)]

N2

∫B(Q)Z(R)\GL2(A)

ϕf (g)ϕE(g)φ2(gf )φ2,R(gR)dg.

Up to this point, we have interpreted the original pairing in Betti cohomology in terms of an integral

of classical automorphic forms, and have translated into adelic language. Notice that ϕf (g) and

φ2(gf ) are right-invariant by T 1(Z), while ϕE(g) is right-invariant under T 2(Z). Let us assume that

ϕf comes from Γ1(N) and generates a representation with central character ω. The above is equal

29

to

C1[GL2(Z) : K(N)]

N2

∫B(Q)Z(R)\GL2(A)

ϕf (g)

∫T 1(Z)

ϕE(gt)dt

φ2(gf )φ2,R(gR)dg

= C1[GL2(Z) : K(N)]

N2

∫B(Q)Z(R)\GL2(A)

ϕf (g)

∫Z(Z)

ϕE(tg)dt

φ2(gf )φ2,R(gR)dg

= C1[GL2(Z) : K(N)]

N2

∫B(Q)Z(R)\GL2(A)

ϕf (g)

∫Z(Z)

ϕE(tg)dt

×

∫Z(Z)

φ2(tgf )ω(t)dt

φ2,R(gR)dg

:= C1[GL2(Z) : K(N)]

N2

∫B(Q)Z(R)\GL2(A)

ϕf (g)ϕE,1(g)φ2,ω−1(gf )φ2,R(gR)dg

= C1[GL2(Z) : K(N)]

N2

∫B(Q)Z(A)\GL2(A)

ϕf (g)ϕE,1(g)φ2,ω−1(gf )φ2,R(gR)dg.

This is exactly the sort of expression to which the automorphic formulation of Rankin’s trick applies.

As in §3.8 of [Bu98], the above is equal to

C1[GL2(Z) : K(N)]

N2

∫A∗×K

Wf (( a 1 )κ)WE,1(( a 1 )κ)(φ2,ω−1φ2,R)(( a 1 )κ)|a|−1dκd∗a.

6.1 Pure tensor decomposition

Finally, we assume that f is a normalized new eigenform on Γ1(N), with nebentypus ω; recall the

parity of ω is equal to the parity of the weight k+ 2. Then the Whittaker function Wf is a product

of local Whittaker functions∏p≤∞Wf,p. In the following section, we will show

Lemma 6.1.1. (i) There exist local functions φ2,ω−1,p such that

φ2,ω−1(( a 1 )κ) =−L(ω,−j − k2 − 1)ω(−1)

N

∏p<∞

φ2,ω−1,p((ap

1 )κp).

We also write φ2,ω−1,∞ = φ2,R.

(ii) There exist local Whittaker functions WE,1,p such that

WE,1(( a 1 )κ) = −ζ(−k1 − j − 1)N

∏p≤∞

WE,1,p((ap

1 )κp).

30

Precise formulae for these local factors will be given in the proof.

Thus

〈ω0, ε π∗(π∗1ρDEisk1+j(φ1) ∪ π∗2ρDEisj+k2(φ2))〉

= C1[GL2(Z) : K(N)]

N4ω(−1)L(ω,−j − k2 − 1)ζ(−k1 − j − 1)

×∫

A∗×K

∏p≤∞

Wf,p((ap

1 )κp)WE,1,p((ap

1 )κp)φ2,ω−1,p((ap

1 )κp)|ap|−1dκ.

We cannot quite exchange the order of product and integration, because the product of the local

integrals may not converge. However, the usual trick of analytic continuation applies. Namely, write

F (s) =∫

A∗×K

∏p≤∞

Wf,p((ap

1 )κp)WE,1,p((ap

1 )κp)φ2,ω−1,p((ap

1 )κp)|ap|s−1dκd∗a.

For s >> 0, we are justified in writing

F (s) =∏p≤∞

∫Q∗p×Kp

Wf,p((ap

1 )κp)WE,1,p((ap

1 )κp)φ2,ω−1,p((ap

1 )κp)|ap|s−1dκpd∗ap

:=∏p≤∞

Ψp(f, s).

In chapter 8 we shall prove the following proposition

Proposition 6.1.1. (i) If p is finite, p - N ,

Ψp(f, s) =Lp(f , 1 + k2 + s)Lp(f , j + k + 2 + s)Lp(ω−1, j + k2 + 2 + 2s)ζp(k1 + j + 2)

.

(ii) Suppose p | N , and that if p = 2 then the local representation $p is not supercuspidal. Then

Ψp(f, s) is a rational function of p−s,and

(p− 1p

)2p+ 1p

Ψp(f, 0) =Lp(f , 1 + k2)Lp(f , j + k + 2)

Lp(ω−1, j + k2 + 2)ζp(k1 + j + 2)

×εp(−j − k

2 −12 , $p, ψp)εp(k1−k2+1

2 , $p, ψp)εp(−j − k2 − 1, ωp, ψp)

1Lp(f,−j)

.

(iii) The factor at infinity is

Ψ∞(f, 0) =ik1+j+2G(j + k + 2)2j+k+2G(k1 + j + 2)

.

31

Granting this, it is clear that F (s) has analytic continuation to all of C, and that

[GL2(Z) : K(N)]N4

F (0) =G(j + k + 2)

2j+k+2G(k1 + j + 2)L(f , 1 + k2)L(f , j + k + 2)

L(ω−1, j + k2 + 2)ζ(k1 + j + 2)

× ik2+k1+j+2

i2kε(−j − k

2 −12 , $)ε(k1−k2+1

2 , $)ω(−1)ε(−j − k2 − 1, ω)

∏p|N

Lp(f,−j)−1.

Recalling that

C1 =2j+k+2πik

2+2k+j2+j−1G(k2 + 1)(j)!G(j + k2 + 2)

,

the value to be computed is

πj!G(k2 + 1)G(j + k + 2)N2G(j + k2 + 2)G(k1 + j + 2)

ω(−1)L(ω,−j − k2 − 1)ζ(−k1 − j − 1)i2k2+j2+2j+k1+1

× L(f , 1 + k2)L(f , j + k + 2)L(ω−1, j + k2 + 2)ζ(k1 + j + 2)

ε(−j − k2 −

12 , $)ε(k1−k2+1

2 , $)ε(−j − k2 − 1, ω)

∏p|N

Lp(f,−j)−1

= (−1)k2+ j2+j

2 (i)k1+j+1πj!G(k1 + 1)G∗(−j)L(f, k1 + 1)L(f,−j)∏p|N

Lp(f,−j)−1

= (−1)k2+ j2+j

2 ik1+1π(2πi)jG(k1 + 1)L(f, k1 + 1)L(N),∗(f,−j).

Finally,

〈δk1 , f〉 =〈(2πi)j∆, ω0〉 = (2πi)j∫∆

ω0 = (2πi)j∞∫0

f(iy)(iy)k1(2πi)dy

= (2πi)j+1ik1G(k1 + 1)L(f, k1 + 1),

and thus the ratio 〈ρD,M∗(1)(ξ),f〉M∗(1)

〈δ,f〉M∗(1)is given by

(−1)k2+ j2+j

2 L(N),∗(f,−j)2

,

, which is precisely the statement of Theorem 4.1.1(i).

32

Chapter 7

Lemmas on Eisenstein Series

The purpose of this chapter is to prove Lemma 6.1.1, and to give more explicit formulae for the

Ψp(f, s). We factor N =∏pnp , and nω will denote the conductor of the Dirichlet character ω. We

will sometimes abuse notation and write ω = ω det : GL2(Z/N)→ C.

7.1 Bernoulli numbers

We wish to give an explicit formula for φ2,ω−1 . Let

κ = (( ap bp

cp dp)p) ∈ GL2(Z)

with reduction κ ∈ GL2(Z/N). With the level N fixed, let γ =∏p|N max(|c|p, |N |p), and write

c′ = γc, N ′ = γN . Recall also that

Af/Z ∼= Q/Z→ [0, 1).

In this we, we view the Bernoulli polynomials as functions on Af . By definition,

φ2,ω−1(κ) =∫

Z∗φ2(κ( 1

x ))ω(x)d∗x

=N j+k2+1

j + k2 + 2

∫Z∗Bj+k2+2(

−cx−1 detκ−1

N)ω(x)d∗x

=N j+k2+1

j + k2 + 2

∫Z∗Bj+k2+2(

−c′x−1 detκ−1

N ′ )ω(x)d∗x

33

=N j+k2+1

j + k2 + 2Icond(ω)|N ′

∏p|N ′

p

p− 1

∫Q

p|N′ Z∗p

Bj+k2+2(−c′xdetκ−1

N ′ )ω−1(x)dx

=N j+k2+1

j + k2 + 2Icond(ω)|N ′

∏p|N ′

p

p− 1ωp(−c′p detκ−1

p )

∫Q

p|N′ Z∗p

Bj+k2+2(x

N ′ )ω−1(x)dx.

For any N , l, the standard distribution relations for Bernoulli numbers may be written as

N l

l

∫Q

p|N Zp

Bl(x

N)ω−1(x)dx = −L(ω,−l + 1).

Using inclusion-exclusion, we obtain that φ2,ω−1(κ) =

− 1N

(N

N ′ )j+k2+2L(ω,−j − k2 − 1)Icond(ω)|N ′

∏p|N ′

p

p− 1ωp(−cp detκ−1

p )Lp(ω,−j − k2 − 1)

.

In other words, if for any Hecke character χ we define a function hχ,p : Zp → C by

hχ,p(x) =

χp(pn)|pn|−j−k2−2 Icondp(χp)=0 if |x|p ≤ |N |p = p−n

pp−1χp(x)|x|

−j−k2−2Lp(χ,−j − k2 − 1)−1 Icondp(χp)+ordpx≤n if |x|p > |N |p

Then

Lemma 7.1.1. With notations as above,

φ2,ω−1(κ) =−L(ω,−j − k2 − 1)ω(−1)

N

∏p

hω,p(cp)ω−1p (κp).

For an arbitrary element gf = (( ap bp

cp dp)p) ∈ GL2(Af ), we chose b = ( b1 ∗

b2) ∈ B(Q), with

bi > 0, and κ = b−1gf ∈ GL2(Z). Set ep = max(|cp|, |dp|), and cp = cpep ∈ Zp. Then |b2|p = ep,

|det b|p = |det gp|p. We defined

φ2,ω−1(gf ) = φ2,ω−1(κ)|b1b2|

j+k2+22

f

34

=−L(ω,−j − k2 − 1)ω(−1)

N| det b(b2)2

|j+k2+2

2f

∏p

hω,p(cp∏l

el)ω−1p (b−1gp)

=−L(ω,−j − k2 − 1)ω(−1)

N

∏p

(|det gp|e2p

)j+k2+2

2 hω,p(cp∏l 6=p

el)ω−1p (gp)

=−L(ω,−j − k2 − 1)ω(−1)

N

∏p

(|det gp|e2p

)j+k2+2

2 hω,p(cp)ω−1p (gp)ωp(

∏l 6=p

el)

=−L(ω,−j − k2 − 1)ω(−1)

N

∏p

(|det gp|e2p

)j+k2+2

2 hω,p(cp)ω−1p (gp)ω−1

p (ep).

Observe that, given a ∈ A∗f , κ ∈ GL2(Z),

φ2,ω−1(( a 1 )κ) =−L(ω,−j − k2 − 1)ω(−1)

N

∏p

|ap|j+k2+2

2p hω,p(cp)ω−1

p (ap)

:=−L(ω,−j − k2 − 1)ω(−1)

N

∏p

φ2,ω−1,p((ap

1 )κp).

This is the expression in (i) of Lemma 6.1.1.

So now we need to calculate out WE,1. As for φ2, extend φ1 to a function φ1 : GL2(Af )→ C by

φ1(gf ) = φ1(b−1gf )(b2b1

)k1+j+2

2 .

Arguing as above, and with the adelic notation there, the same computation yields

φ1,1(gf ) =−ζ(−k1 − j − 1)

N

∏p

(|det gp|e2p

)k1+j+2

2 h1,p(dp)

:=−ζ(−k1 − j − 1)

N

∏p

φ1,1,p(gp).

For good measure, we shall review how to calculate the Whittaker function attached to an

Eisenstein series. Recalling that ±U(Z)\SL2(Z) ∼= B(Q)\GL2(Q), one can check that

∑γ∈±U(Z)\SL2(Z)

φ1(γκ)j(γ, τ)k1+j+2

=∑

B(Q)\GL2(Q)

φ1(γκ)(det γ)k1+j+2

2

j(γ, τ)k1+j+2

35

Given g∞ ∈ GL2(R), write again c = ( 1sgn(det g∞) ), g′∞ = cg∞ and b′ = cbc. Then

φE(gf , g∞) = FE(b′−1g′∞, cb−1gf )

= Eφ1(b′−1g′∞, cb

−1gf )Im(b′−1g∞ · i)k1+j+2

2 ϑ(b′−1g′∞)k1+j+2

2

=∑

B(Q)\GL2(Q)

φ1(γcb−1gf )(det γ)k1+j+2

2

j(γ, b′−1g′∞)k1+j+2(b2b1

)k1+j+2

2 Im(g∞ · i)k1+j+2

2 ϑ(g′∞)k1+j+2

2

= ±∑

B(Q)\GL2(Q)

φ1(γgf )(det γ)k1+j+2

2

j(γc, g′∞)k1+j+2Im(g∞ · i)

k1+j+22 ϑ(g′∞)k1+j+2.

A complete set of coset representatives for B(Q)\GL2(Q) are given by ( 11 ) and

( −11 λ

), λ ∈ Q,

so

WE(g∞, gf ) =∫

U(Q)\U(A)

φE(m∞g∞,mfgf )ψ(m)−1dm

=∫

U(Q)\U(A)

∑B(Q)\GL2(Q)

φ1(γmfgf )(det γ)k1+j+2

2

j(γc, cm∞g∞)k1+j+2

× Im(cm∞g∞ · i)k1+j+2

2 ϑ(cm∞g∞)k1+j+2ψ(m)−1dm

=[ ∫U(Q)\U(A)

∑B(Q)\GL2(Q)

φ1(γmfgf )(det γ)k1+j+2

2

j(γm∞c, g′∞)k1+j+2ψ(m)−1dm

]

× Im(g′∞ · i)k1+j+2

2 ϑ(g′∞)k1+j+2.

The ( 11 ) coset does not contribute to the integral, so we get

∫U(Q)\U(A)

∑U(Q)

φ1(w0γmfgf )j(w0γm∞c, g′∞)k1+j+2

ψ(m)−1dm

Im(g′∞ · i)k1+j+2

2 ϑ(g′∞)k1+j+2

=

∫U(A)

φ1(w0mfgf )j(w0m∞c, g′∞)k1+j+2

ψ(m)−1dm

Im(g′∞ · i)k1+j+2

2 ϑ(g′∞)k1+j+2

=[Im(g′∞ · i)

k1+j+22 ϑ(g′∞)k1+j+2

∫U(R)

ψR(mR)−1dmR

j(w0m∞c, g′∞)k1+j+2

]

×

∫U(Af )

φ1(w0mfgf )ψf (mf )−1dmf

.

36

Writing WE,∞ for the term at the infinite prime, one has immediately

WE,1(g∞, gf ) = WE,∞(g∞)∫Af

φ1,1(( −11 x )gf )ψf (x)−1dx

=−ζ(−k1 − j − 1)

NWE,∞(g∞)

∏p

∫Qp

φ1,1,p(( −11 x )gp)ψf (x)−1dx

:=−ζ(−k1 − j − 1)

N

∏p≤∞

WE,1,p(gp).

Thus Proposition 6.1.1 is proven.

7.2 An explicit kernel

The rest of the chapter is devoted to making more explicit the integrals

Ψp(f, s) =∫

Q∗p×Kp

Wf,p((ap

1 )κp)WE,1,p((ap

1 )κp)φ2,ω−1,p((ap

1 )κp)|ap|s−1dκ,

at least for p | N . With p fixed, n = np denotes the exponent of p in N , and | · | = | · |p. In this

situation, the product Wf,pφ2,ω−1,p is invariant under right translation by K0(pn), so

Ψp(f, s) =∫

Q∗p×GL2(Zp)/K0(pn)

Wf,p(( a 1 )κ)φ2,ω−1,p(( a 1 )κ)|a|s−1p

×∫

K0(pn)

WE,1,p(( a 1 )κγ)dγdκd∗a.

Let us write

K(a, κ, s) = φ2,ω−1,p(( a 1 )κ)|a|s−1p

∫K0(pn)

WE,1,p(( a 1 )κγ)

for the kernel appearing, so that

Ψp(f, s) =∫

Q∗p×GL2(Zp)/K0(pn)

Wf,p(( a 1 )κ)K(a, κ, s)dκd∗a. (7.1)

Now a complete set of coset representatives for GL2(Zp)/K0(pn) is given by

( x −11 ), 0 < x ≤ pn, ( 1

x 1 ), 0 < x ≤ pn, p | x.

37

Lemma 7.2.1. For 0 < x ≤ pn,

K(a, ( x −11 ), s) = ψ(−ax)|a|s+

k2−k12 (

p

p− 1)2

ζp(k1 + j + 1)Lp(ω,−j − k2 − 1)ζp(k1 + j + 2)

× IZp(pa)ω−1(a)

[1− pk1+j − p− 1

p|pa|k1+j+1

].

For 0 < x < pn, p | x,

K(a, ( 1x 1 ), s) = |a|s+j+ k

2 +1(p

p− 1)2

1Lp(ω,−j − k2 − 1)ζp(k1 + j + 2)

IZp(a)

× I|x||nω|≥p−nω(xa−1)|x|−j−k2−2

and

K(a, ( 11 ), s) = |a|s+j+ k

2 +1 p

p− 11

ζp(k1 + j + 2)IZp

(a)I|nω|=1ω(pna−1)|pn|−j−k2−2.

Combining this lemma with the known formulae for normalized newforms Wf , the equation (7.1)

is completely explicit, and will be evaluated in chapter 8.

Proof.

K(a, κ, s) = φ2,ω−1,p(( a 1 )κ)|a|s−1p

∫K0(pn)

WE,1,p(( a 1 )κγ)dγ.

From above, for 0 ≤ x < pn, 0 < y < pn

φ2,ω−1,p(( a 1 )( x −11 )) = |a|

j+k2+22

p

p− 1Lp(ω,−j − k2 − 1)−1ω−1(a)

φ2,ω−1,p(( a 1 )( 1y 1 )) = |a|

j+k2+22 I|ynω|≥p−nω(ya−1)|y|−j−k2−2 p

p− 1Lp(ω,−j − k2 − 1)−1

φ2,ω−1,p( a 1 ) = |a|j+k2+2

2 I|nω|=1ω(pna−1)|pn|−j−k2−2.

So we just need to calculate

∫K0(pn)

WE,1,p(( a 1 )( x −11 )γ)dγ

and ∫K0(pn)

WE,1,p(( a 1 )( 1x 1 )γ)dγ.

First, a quick calculation yields

38

Lemma 7.2.2. For g = ( a bc d ), p | N ,

1vol(K0(pn))

∫K0(pn)

φ1,1,p(gγ)dγ =

1 c ∈ Z∗ppp−1ζp(−k1 − j − 1)−1 c /∈ Z∗p

Now, for the former,

ψ(ax)−1|a|k1+j

2

∫K0(pn)

WE,1,p(( a 1 )( x −11 )γ)dγ

= ψ(ax)−1

∫K0(pn)

∫Qp

φ1,1,p(( −11 y )( x −1

1 )γ)ψ(ay)dydγ

=∫

K0(pn)

∫Qp

φ1,1,p(( 1y 1 )γ)ψ(ay)dydγ

=∫

K0(pn)

[ ∫pZp

φ1,1,p(( 1y 1 )γ)ψ(ay)dy

+∫

Qp−pZp

φ1,1,p(( ∗ ∗1 ∗ )γ)|y|−(k1+j+2)ψ(ay)dy]dγ

=∫pZp

[ ∫K0(pn)

φ1,1,p(( 1y 1 )γ)dγ

]ψ(ay)dy

+∫

Qp−pZp

∫K0(pn)

φ1,1,p(( ∗ ∗1 ∗ )γ)dγ

|y|−(k1+j+2)ψ(ay)dy

=∫pZp

p

p− 1ζp(−k1 − j − 1)−1ψp(ay)dy +

∫Qp−pZp

|y|−(k1+j+2)ψ(ay)dy

=ζp(−k1 − j − 1)−1

p− 1IZp

(pa) +0∑

m=−∞|p|−m(k1+j+1)

∫Z∗p

ψ(apmy)dy

=ζp(−k1 − j − 1)−1

p− 1IZp

(pa) +0∑

m=−∞|p|−m(k1+j+1)

[IZp

(pma)− 1p

IZp(pm+1a)

]

=ζp(−k1 − j − 1)−1

p− 1IZp(pa) +

0∑m=−orda

|p|−m(k1+j+1) − 1p

0∑m=−orda−1

|p|−m(k1+j+1)

= IZp(pa)

[ζp(−k1 − j − 1)−1

p− 1+

1− |p|(orda+1)(k1+j+1) − 1p + 1

p |p|(orda+2)(k1+j+2)

1− p−k1−j−1

]

= IZp(pa)[p(1− pk1+j)(1− p−k1−j−2)

(p− 1)(1− p−(k1+j+1))− |pa|

k1+j+1(1− p−k1−j−2)1− p−k1−j−1

],

39

as claimed. Similarly, for x ∈ pZp,

|a|k1+j

2

∫K0(pn)

WE,1,p(( a 1 )( 1x 1 )γ)dγ

=∫

K0(pn)

∫Qp

φ1,1,p(( −11 y )( 1

x 1 )γ)ψ(ay)dydγ

=∫

K0(pn)

∫Qp

φ1,1,p(( −x −11+xy y )γ)ψ(ay)dydγ

=∫Zp

[ ∫K0(pn)

φ1,1,p(( ∗ ∗1 ∗ )γ)dγ]ψ(ay)dy

+∫

Qp−Zp

∫K0(pn)

φ1,1,p((∗ ∗

y−1+x ∗ )γ)dγ

|y|−k1−j−2ψ(ay)dy

=∫Zp

ψ(ay)dy +∫

Qp−Zp

p

p− 1ζp(−k1 − j − 1)−1|y|−k1−j−2ψ(ay)dy

= IZp(a) +p

p− 1ζp(−k1 − j − 1)−1

−1∑m=−∞

|p|−m(k1+j+1)

∫Z∗p

ψp(apmy)dy

= IZp(a)|a|k1+j+1(1− p−k1−j−2)

p

p− 1.

40

Chapter 8

The Local Computations

In this chapter we prove Proposition 6.1.1. The computations 6.1.1(i) and 6.1.1(iii) (which do not

depend on the choice of residue data φi) are well-known:

8.1 Classical calculations

8.1.1 The unramified computation

Suppose now that the local representation $p is unramified. Then the integral in question is merely

∫Q∗p

Wf ( a 1 )WE,1( a 1 )|a|s+j+k2

2 ω−1(a)d∗a.

This may be evaluated as in, for example, Lemma 15.9.4 of [Ja72], and one obtains

Lp($p ⊗ ω−1, s+ k2−k12 )Lp($p ⊗ ω−1, s+ j + k

2 )Lp(ω−1, j + k2 + 2 + 2s)

=Lp($p, s+ k2−k1+1

2 )Lp($p, s+ j + k2 + 32 )

Lp(ω−1, j + k2 + 2 + 2s)

=Lp(f , 1 + k2 + s)Lp(f , j + k + 2 + s)

Lp(ω−1, j + k2 + 2 + 2s).

8.1.2 Computation at infinity

The integral to be computed is Ψ∞(f, s) =

∫R∗×SO(2)

Wf,∞(( a 1 )θ)WE,1,∞(( a 1 )θ)φ2,ω−1,∞(( a 1 )θ)|a|s−1dθd∗a.

41

A formula for Wf,∞ can be found in [Bu98], §2.8, although one must modify for our choice of ψ∞.

One finds

Wf,∞(( a 1 )θ) = IR∗+(a)ak+22 e−2πaθk+2.

Along with the formula for φ2,R and WE,∞,

Ψ∞(f, s) =∫

R∗+×SO(2)

ak+22 e−2πa

a k1+j+22

∫R

ψ∞(x)−1dx

(ai+ x)k1+j+2

a j+k2+22 as−1dθd∗a

=

∞∫0

ak+j+s+2e−4πa

∫R+ai

ψ∞(x)−1dx

xk1+j+2d∗a

=

∞∫0

ak+j+s+2e−4πa −ik1+j+2

G(k1 + j + 2)d∗a

=ik1+j+2

G(k1 + j + 2)(4π)k+j+s+2

∞∫0

ak+j+s+2e−ad∗a

=ik1+j+2

G(k1 + j + 2)(4π)k+j+s+2Γ(k + j + s+ 2).

This function is clearly analytic in s, and taking s = 0 gives the asserted value.

8.2 The bad integrals

We are left to do the bad local zeta integrals of 6.1.1(ii). Fix a prime p | N , and set n = ordpN . It

will also be convenient to change notation and write nω for ordp(condω). Then

Ψp(f, s) =∫

Q∗p×GL2(Zp)/K0(pn)

Wf (( a 1 )κ)K(a, κ, s)dκd∗a,

where K is the kernel given in Lemma 7.2.1. It is well-known that such integrals give rational

functions of p−s. Note also that there is an analogous definition of K for larger p-adic fields, and in

fact the computations that follow make no special use of the fact that the ground field is Q.

A complete set of left cosets for K0(pn) in GL2(Zp) is given by ( x −11 ), 1 ≤ x ≤ pn and

( 1px 1 ), 1 ≤ x ≤ pn−1. The latter set stratifies naturally as ∪nl=1Sl, Sl = ( 1

x 1 ), 1 ≤ x ≤

pn, ordp(x) = l, and it is further natural to write S0 = ( x −11 ), 1 ≤ x ≤ pn. Thus the inte-

gral in question may be written

n∑l=0

∞∑m=−∞

∫Sl

∫pmZ∗p

Wf (( a 1 )κ)K((a, κ, s)d∗adκ.

42

The volume normalizations are such that∫Zp

dx =∫Z∗pd∗x = 1, and, given any function H on

GL2(Zp)/Γ0(pn), ∫S0

H(κ)dκ =p

p+ 1

∫Zp

H( x −11 )dx

∫Sn

H(κ)dκ =p

p+ 1p

p− 1

∫pnZ∗p

H( 1x 1 )dx =

p

(p+ 1)pnH( 1

1 ),

and, for 0 < l < n, ∫Sl

H(κ)dκ =p

p+ 1

∫plZ∗p

H( 1x 1 )dx.

8.2.1 Supercuspidal case

Suppose $p is supercuspidal, and restrict to the situation where p is odd. This means (e.g., [Sc02])

that one is given a quadratic extension E of F = Qp and a character ξ : E∗ → C∗ that does not

factor through NE/F . The contragredient representation $p is defined by using ξ−1. We fix the

additive character ψF = ψp from above, ψE = ψF TrE/F , and let χE/F : F ∗ → C∗ be the quadratic

character attached to the extension E/F .

The Weil representation ΩE of SL2(F ) on C∞c (E) is given by

(ΩE( 1 x1 )f)(v) = ψF (xNE/F (v))f(v)

(ΩE( a a−1 )f)(v) = |a|FχE/F (a)f(av)

(ΩE( 1−1 )f)(v) = γ

∫E

f(u)ψE(uv)du

where, according to [JL70], Theorem 4.6 and earlier,

γ =εp(s, $p, ψp)εp(s, ξ−1, ψE)

γ2 = χE/F (−1).

Set Uξ ⊂ C∞c (E) to be the set of functions f such that

f(yv) = ξ(y)−1f(v), y ∈ ker(NE/F ), v ∈ E.

The action of SL2(F ) on Uξ extends to a representation Ωξ,χFof GL+

2 (F ) (the set of matrices with

determinant in NE/FE∗) by

(Ωξ,ψF( a 1 )f)(v) = |a|

12F ξ(b)f(bv), NE/F b = a.

43

The representation Ωξ of GL2(F ) obtained by induction is the dihedral supercuspidal $p, and the

vector space of functions f : GL2(F )→ Uξ on which it acts will be denoted Vξ. The central character

of Ωξ is known to be χE/F ξ|F∗ . Let φ0 ∈ Uξ be given by

φ0(v) = ξ−1(v)IO∗E (v).

Then the normalized newvector in the model Vξ is given as

Proposition 8.2.1. ([Sc02], Theorem 2.3.5) If E/F is unramified, the normalized

newvector in the model Vξ of Ωξ is defined by

f(g) =

Ωξ,ψF(g)φ0 if g ∈ GL+

2 (F )

0 if g /∈ GL+2 (F ).

If E/F is ramified, choose a unit x ∈ O∗F −NE/FO∗E. The normalized newvector in the model Vξ of

Ωξ is

f(g) =

Ωξ,ψF(g)φ0 if g ∈ GL+

2 (F )

Ωξ,ψF(g( x 1 ))φ0 if g /∈ GL+

2 (F ).

A Whittaker functional Vξ → C is given by

ΛψF(h) = h( 1

1 )(1E),

and the normalized Whittaker function for $p = Ωξ is thus

Wf (g) = f(g)(1E).

Recall that the goal is to calculate

Ψp(f, s) =n∑l=0

∞∑m=−∞

∫Sl

∫pmZ∗p

Wf (( a 1 )κ)K(a, κ, s)d∗adκ.

For κ ∈ S0, K(( a 1 )κ) has support on a ∈ p−1Zp, while Wf (( a 1 )κ) has support on a ∈ p−nZ∗p.

Since n ≥ 2, these supports are disjoint, and the large stratum S0 does not contribute to the integral.

44

Hence

(p− 1p

)2p+ 1p

Ψp(f, s)

= (p− 1p

)2n−1∑l=1

∞∑m=−∞

∫plZ∗p

∫pmZ∗p

Wf (( a 1 )( 1x 1 ))K(a, ( 1

x 1 ), s)d∗adx

+ (p− 1p

)2|p|n∞∑

m=−∞

∫pmZ∗p

Wf ( a 1 )K(a, ( 11 ), s)d∗adx

=n−nω∑l=1

∑m≥0

Cl,m

∫plZ∗p

∫pmZ∗p

Wf (( a 1 )( 1x 1 ))ω(xa−1)d∗adx

where for l < n,

Cl,m =|p|−l(j+k2+2)|p|m(s+j+ k

2 +2)

ζp(k1 + j + 2)Lp(ω,−j − k2 − 1)

while

Cn,m =|p|−n(j+k2+2)|p|m(s+j+ k

2 +2)

ζp(k1 + j + 2).

By elementary algebra, Proposition 6.1.1(ii) in this case follows immediately from

Lemma 8.2.1. If the central character ω is ramified, then for 1 ≤ l ≤ n− nω, 0 ≤ m,

Ψp,l,m(f, s) =∫plZ∗p

∫pmZ∗p

Wf (( a 1 )( 1x 1 ))ω(xa−1)d∗adx

vanishes unless l = n− nω, m = 0, in which case it is equal to

ε(1, $p, ψF )2

ε(1, ω, ψF ).

If ω is unramified,

Ψp,l,m(f, s) =

εp(1, $p, ψF )2 p−1

p if l = n,m = 0

−εp(1, $p, ψF )2ω(p)−1 if l = n− 1,m = 0

0 else.

Proof. For uniformity of notation, if E/F is ramified, we will write p12O∗E = pkOE − pk+1OE . As

another bit of notation, given a ∈ NE∗ ⊂ F ∗, we will write b = b(a) for any choice of b ∈ E∗ with

Nb = a. We may choose b to be a locally continuous function of a.

45

First, suppose E/F is ramified. Choose a t ∈ F\NE∗ of valuation 1. Then

∫plZ∗p

∫pmZ∗p

Wf (( a 1 )( 1x 1 ))ω(xa−1)d∗adx

=∫plZ∗p

∫pmZ∗p

Wf ((−1−1 )( a 1 )( −1

1 )( 1 −x1 )( −1

1 ))ω(xa−1)d∗adx

=∫plZ∗p

∫pmZ∗p∩NE∗

Ωξ,ψf((−1

−1 )( a 1 )( −11 )( 1 −x

1 )( −11 ))φ0(1)ω(xa−1)d∗adx

+∫plZ∗p

∫pmZ∗p∩(F∗−NE∗)

Ωξ,ψF((−1

−1 )( at−1

1)( −1

1 )( 1 −xt−1

1)( −1

1 ))φ0(1)ω(xa−1)d∗adx

= 2∫plZ∗p

∫pmZ∗p∩NE∗

Ωξ,ψF((−1

−1 )( a 1 )( −11 )( 1 −x

1 )( −11 ))φ0(1)ω(xa−1)d∗adx

= 2∫plZp

∫pmZp∩NE∗

∫E

∫OE

ξ(bv−1)ψF (−xN(u))ψE(uv)ψE(−ub)ω(xa−1)dv

|v|dud∗adx.

Replacing x 7→ xa, v 7→ vb, u 7→ ub−1, this is equal to

2∫

p(l−m)Z∗p

∫pmZ∗p∩NE∗

∫E

∫p−

m2 O∗E

ξ(v−1)ψF (−xN(u))ψE(uv)ψE(−u)ω(x)dv

|v|dud∗adx

=∫

p(l−m)Z∗p

∫E

∫p−

m2 O∗E

ξ(v−1)ψF (−xN(u))ψE(uv)ψE(−u)ω(x)dv

|v|dudx

= |pl−m|∞∑

t=−∞

[ ∫ptO∗E

∫p−

m2 O∗E

ξ(v−1)ω(−N(u)−1)ψE(uv)ψE(−u)dv|v|du

] ∫pl−m−tZ∗p

ψF (x)ω(x)dx

|x|

= |pl−m|γ2∞∑

t=−∞

∫ptO∗E

∫pt−mO∗E

ξ−1(uv)ψE(v)ψE(u)dv

|v|du

∫pl−m−tZ∗p

ψF (x)ω(x)dx

|x|

The first integral vanishes unless t = −n,m = 0, so

Ψp,l,m(f, s) = |p|l−nγ2ε(1, ξ, ψE)2∫

pl−nZ∗p

ω(x)ψF (x)dx

|x|

= ε(1, $p, ψE)2∫

pl−nZ∗p

ω(x)ψF (x)dx.

On the other hand, if E/F is unramified, the same basic calculation yields

∫plZ∗p

∫pmZ∗p

Wf (( a 1 )( 1x 1 ))ω(xa−1)d∗adx =

46

Im even

∫p(l−m)Z∗p

∫E

∫p−

m2 O∗E

ξ(v−1)ψF (−xN(u))ψE(uv)ψE(−u)ω(x)dv

|v|dudx,

and one may proceed in the same manner.

8.2.2 Principal series or special with two ramified characters

Suppose$p∼= π(χ1, χ2) is principal series, where both χ1 and χ2 are ramified, or that$p

∼= σ(χ1, χ2)

is special with ramified characters. The Whittaker functions, L-, and ε-factors are given by identical

formulae ([Sc02], pp. 126, 130), so we can assume $p is principal series. Write ni > 0 for the

conductor of χi, so that n = n1 + n2. We may take n1 ≥ n2. The central character is ω = χ1χ2, so

nω ≤ n1, with equality unless n1 = n2.

We need to compute

Ψp(f, s) =n∑l=0

∞∑m=−∞

∫Sl

∫pmZ∗p

Wf (( a 1 )κ)K(( a 1 )κ)d∗adκ.

For κ ∈ S0, K(( a 1 )κ) has support on a ∈ p−1Zp, while Wf (( a 1 )κ) has support on a ∈ p−nZ∗p.

Since n ≥ 2, these supports are disjoint, and the large stratum S0 does not contribute to the integral.

Thus, as in the supercuspidal case, and with the same constants,

(p− 1p

)2p+ 1p

Ψp(f, s) =n−nω∑l=1

∑m≥0

Cl,m

∫plZ∗p

∫pmZ∗p

Wf (( a 1 )( 1x 1 ))ω(xa−1)d∗adx,

and Proposition 6.1.1(ii) will again follow from

Lemma 8.2.2. If the central character ω is ramified, then for 1 ≤ l ≤ n− nω, 0 ≤ m, Ψp,l,m(f, s)

vanishes unless l = n− nω, m = 0, in which case it is equal to

ε(1, $p, ψp)2

ε(1, ω, ψp)

If ω is unramified,

Ψp,l,m(f, s) =

εp(1, $p, ψF )2 p−1

p if l = n,m = 0

−εp(1, $p, ψF )2ω(p)−1 if l = n− 1,m = 0

0 else.

47

In the standard model of π(χ1, χ2), one K1(pn)-invariant vector is given by

g0( 1x 1 ) = χ1(x)−1Ipn2Z∗p(x),

and the intertwining map from the standard model of π(χ1, χ2) to the space of Whittaker functions

is

g 7→Wg : h→∫Qp

g(( −11 y )h)ψp(y)dy.

For our g0,

Wg0( 11 ) = εp(1, χ−1

2 , ψp).

Thus

Wf (( a 1 )( 1x 1 )) =

1εp(1, χ−1

2 , ψp)

∫Qp

g(( −11 y )( a 1 )( 1

x 1 ))ψp(y)dy

=|a|

εp(1, χ−12 , ψp)

∫Qp

g(( y−1 1ay )( 1

y−1+x 1 ))ψp(ay)dy

=χ2(a)|a|

12

εp(1, χ−12 , ψp)

∫Qp

χ2(y)χ1(1 + yx)−1Ipn2Z∗p(y−1 + x)ψp(ay)dy

|y|.

We may compute as follows:

Proof. For 1 ≤ l ≤ n− nω,

∫plZ∗p

∫pmZ∗p

Wf (( a 1 )( 1x 1 ))ω(xa−1)d∗adx

=∫plZ∗p

∫pmZ∗p

∫Qp

χ2(a)|a|12

εp(1, χ−12 , ψp)

χ2(y)χ1(1 + yx)−1

Ipn2Z∗p(y−1 + x)ψp(ay)ω(xa−1)dy

|y|d∗adx

=|p|m2

εp(1, χ−12 , ψp)

∫plZ∗p

∫pmZ∗p

∫Qp

χ2(ayx−1)χ1(1 + y)−1ψp(ayx−1)

Ipn2Z∗p(x(y−1 + 1))ω(xa−1)dy

|y|d∗adx

48

=|p|m2 |pl|

εp(1, χ−12 , ψp)

∫pmZ∗p

∫Qp

χ2(ay)χ1(1 + y)−1ω(a)−1

Ipn2−lZ∗p(y−1 + 1)∫plZ∗p

χ−12 (x)ω(x)ψp(ayx−1)

dx

|x|dy

|y|d∗a

=|pl+ m

2 |εp(1, χ−1

2 , ψp)

∫pmZ∗p

∫Qp

χ2(ay)χ1(1 + y)−1ω(a)−1

Ipn2−lZ∗p(y−1 + 1)∫

p−l+ordy+mZ∗p

χ1(x−1ay)ψp(x)dx

|x|dy

|y|d∗a

=|pl+ m

2 |ε(1, χ1, ψp)εp(1, χ−1

2 , ψp)

∫pmZ∗p

∫pl−n1−mZ∗p

ω(y)χ1(1 + y)−1Ipn2−lZ∗p(y−1 + 1)dy

|y|d∗a

=|pl+ m

2 |ε(1, χ1, ψp)εp(1, χ−1

2 , ψp)

∫pm+n1−lZ∗p

ω−1(y)χ1(1 + y−1)−1Ipn2−lZ∗p(y + 1)dy

|y|

=|pl+ m

2 |ε(1, χ1, ψp)εp(1, χ−1

2 , ψp)

∫pm+n1−lZ∗p

χ−12 (y)χ1(1 + y)−1Ipn2−lZ∗p(y + 1)

dy

|y|.

If m + n1 > n2, the integrand vanishes unless l = n2, in which case we get (writing t =

m+ n1 − n2 > 0)

|pn2+m2 |ε(1, χ1, ψp)

εp(1, χ−12 , ψp)

∫pt

χ−12 (y)χ−1

1 (1 + y)dy

|y|

=|pn2+

m2 |ε(1, χ1, ψp)

εp(1, χ−12 , ψp)

∫p−t

ω(y)χ−11 (1 + y)

dy

|y|.

And

∫p−tZ∗p

ω(y)χ−11 (1 + y)

dy

|y|

∫p−nω Z∗p

ω−1(δ)ψp(δ)dδ

|δ|

=∫

p−tZ∗p

∫p−nω+tZ∗p

ω−1(δ)χ−11 (1 + y)ψp(δy)

dδ

|δ|dy

|y|

=∫

p−tZ∗p

∫p−nω+tZ∗p

ω−1(δ)χ−11 (y)ψp(δy − δ)

dδ

|δ|dy

|y|

=∫

p−nχ Z∗p

χ−11 (y)ψp(y)

dy

|y|

∫p−nω+tZ∗p

χ−12 (δ)ψp(−δ)

dδ

|δ|.

This last expression vanishes unless nω = n1 and t = nχ − n2, so in this case Ψp,l,m = 0 unless

49

l = n2, m = 0, where we get

εp(1, χ1, ψp)εp(1, χ2, ψp)εp(1, ω, ψp)

=εp(1, $p, ψp)εp(1, ω, ψp)

.

The other possibility is that m = 0 and n1 = n2, when the integral

∫pn1−lZ∗p

χ−12 (y)χ−1

1 (1 + y)Ipn1−lZ∗p(y + 1)dy

|y|

clearly vanishes unless l ≥ n1. If l > n1, we must evaluate

∫pn1−lZ∗p

χ−12 (y)χ−1

1 (1 + y)dy

|y|.

Multiplying by εp(1, χ2, ψp) and rearranging integrals as above, we obtain

Ψp,l,m = Im=0εp(1, $p, ψp)2∫

pl−nZ∗p

ω(y)ψp(y)dy,

which gives the desired answer. Finally, we must consider m = 0, l = n1 = n2, where the integral is

∫Z∗p

χ−12 (y)χ−1

1 (1 + y)IZ∗p(1 + y)dy

|y|.

Comparing conductors, ∫pZp−1

χ−12 (y)χ−1

1 (1 + y)dy = 0,

so that ∫Z∗p

χ−12 (y)χ−1

1 (1 + y)IZ∗p(1 + y)dy

|y|=∫Z∗p

χ−12 (y)χ−1

1 (1 + y)dy

|y|,

and the calculation proceeds as above.

8.2.3 Principal series with one ramified character

Suppose $p∼= π(χ1, χ2), with χ1 ramified (of conductor n) and χ2 unramified. As usual, our goal

is to compute

Ψp(f, s) =n∑l=0

∞∑m=−∞

∫Sl

∫pmZ∗p

Wf (( a 1 )κ)K(a, κ, s)d∗adκ.

50

A newvector in the standard model for $p is determined by

g( 1x 1 ) = χ−1

1 (x)χ2(x)|x|−1Iordx≤0,

and the Whittaker function attached to g is normalized. Since nω = n, K(a, κ, s) is supported on

κ ∈ S0; recalling the formula for K,

(p− 1p

)2p+ 1p

Ψp(f, s)

= |p|n ζp(k1 + j + 1)ζp(k1 + j + 2)

∞∑m=−1

pn−1∑x=0

∫pmZ∗p

Wf (( a 1 )( x 11 ))ψ(−ax)

× |a|s+k2−k1

2 ω−1(a)[1− pk1+j − p− 1

p|pa|k1+j+1

]d∗a

=ζp(k1 + j + 1)ζp(k1 + j + 2)

∞∑m=−1

|p|m(s+k2−k1

2 )

[1− pk1+j − p− 1

p|p|(m+1)(k1+j+1)

]×∫

pmZ∗p

Wf (( a−1 )ω−1(a)d∗a.

Easily,

∫pmZ∗p

Wf (( a−1 )ω−1(a)d∗a = Wf (

1−p−m )

= |p|m2∫Qp

χ1(p−m)g( 1y 1 )ψ(p−my)dy

= |p|m2∫

Qp−Zp

χ−11 (pmy)χ2(y)ψ(p−my)

dy

|y|

= |p|m2 χ2(p−m)∫

Qp−pmZp

χ−11 (y)χ2(y)ψ(y)

dy

|y|

= |p|m2 χ2(p−m)∫

p−n1

χ−11 (y)χ2(y)ψ(y)

dy

|y|

= |p|m2 χ2(p−m)εp(1, χ1, ψp)2

εp(1, ω, ψp),

51

and so

(p− 1p

)2p+ 1p

Ψp(f, s)

=ζp(k1 + j + 1)ζp(k1 + j + 2)

εp(1, χ1, ψp)2

εp(1, ω, ψp)

×∞∑

m=−1

χ2(p−m)|p|m(s+k2−k1

2 + 12 )

[1− pk1+j − p− 1

p|p|(m+1)(k1+j+1)

]=ζp(k1 + j + 1)ζp(k1 + j + 2)

εp(1, χ1, ψp)2

εp(1, ω, ψp)|p|−(s+

k2−k12 + 1

2 )

×

[1− pk1+j

1− χ−12 (p)p−(s+

k2−k12 + 1

2 )− (p− 1)/p

1− χ−12 (p)p−(s+j+ k

2 + 32 )

]

=ζp(k1 + j + 1)ζp(k1 + j + 2)

εp(1, χ1, ψp)2

εp(1, ω, ψp)|p|−(s+

k2−k12 + 1

2 )Lp(χ−12 , s+

k2 − k1

2+

12)

× Lp(χ−12 , s+ j +

k

2+

32)(1− pk1+j+1)

p(1− χ−1

2 (p)p−(s+j+ k2 + 1

2 ))

=Lp(χ−1

2 , s+ k2−k12 + 1

2 )Lp(χ−12 , s+ j + k

2 + 32 )

ζp(k1 + j + 2)εp(1, χ1, ψp)2

εp(1, ω, ψp)1

Lp(χ2,−s− j − k2 −

12 )

=Lp(f , s+ k2 + 1)Lp(f , s+ j + k + 2)

ζp(k1 + j + 2)εp(1, $p, ψp)2

εp(1, ω, ψp)1

Lp(f,−s− j).

8.2.4 Special with conductor p

The only case remaining is where $p is the special representation σ(χ| · | 12 , χ| · |− 12 ) for an unramified

character χ. This representation has conductor 1, and central character ω = χ2. We have

ε(s,$p, ψp) = −χ−1(p)p12−s.

The normalized Whittaker function for $p is given by

Wf ( a 1 ) = χ(a)|a|IZp(a)

Wf ( a−1 ) = −χ(a)|pa|IZp(pa).

52

Combining this with the formulas for K given above,

(p− 1p

)2p+ 1p

Ψp(f, s)

= (p− 1p

)21∑l=0

∞∑m=−∞

∫Sl

∫pmZ∗p

Wf (( a 1 )κ)K(a, κ, s)d∗adκ

= (p− 1p

)2[p−1

∞∑m=0

∫pmZ∗p

χ(a)|a|K(a, ( 11 ), s)d∗a

+∞∑

m=−1

∫pmZ∗p

−χ(a)|pa|K(a, ( 1−1 ), s)d∗a

]

:= I1(s) + I0(s)

We compute

I1(s) = |p|∞∑m=0

∫pmZ∗p

χ(a)|a|s+j+ k2 +2 p− 1

p

1ζp(k1 + j + 2)

ω(pa−1)|p|−j−k2−2

= ω(p)p− 1p

1ζp(k1 + j + 2)

|p|−j−k2−1∞∑m=0

χ−1(pm)|pm|s+j+ k2 +2

= ω(p)p− 1p

1ζp(k1 + j + 2)

|p|−j−k2−1Lp(χ−1, s+ j +k

2+ 2)

and

I0(s) = −∞∑

m=−1

∫pmZ∗p

χ(a)|pa||a|s+k2−k1

2ζp(k1 + j + 1)

Lp(ω,−j − k2 − 1)ζp(k1 + j + 2)

× ω−1(a)[1− pk1+j − p− 1

p|pa|k1+j+1

]d∗a

= −|p|−(s+k2−k1

2 ) ζp(k1 + j + 1)Lp(ω,−j − k2 − 1)ζp(k1 + j + 2)

χ(p)

×∞∑m=0

χ−1(pm)|pm|s+k2−k1

2 +1

[1− pk1+j − p− 1

p|a|k1+j+1

]d∗a

= −|p|−(s+k2−k1

2 ) ζp(k1 + j + 1)Lp(ω,−j − k2 − 1)ζp(k1 + j + 2)

χ(p)

×

[1− pk1+j

1− χ−1(p)p−(s+k2−k1

2 +1)− 1− p−1

1− χ−1(p)p−(2+j k

2+2)

]

= −|p|−(s+k2−k1

2 ) ζp(k1 + j + 1)Lp(ω,−j − k2 − 1)ζp(k1 + j + 2)

χ(p)(1− pk1+j+1)

p

× Lp(χ−1, s+k2 − k1

2+ 1)Lp(χ−1, s+ j +

k

2+ 2)(1− χ−1(p)p−(s+j+ k

2 +1))

= −|p|Lp(χ−1, s+ k2−k1

2 + 1)Lp(χ−1, s+ j + k2 + 2)

Lp(ω,−j − k2 − 1)ζp(k1 + j + 2)(1− χ(p)ps+j+

k2 +1).

53

So

Ψp(f, 0) = I1(0) + I0(0)

= ω(p)|p|−j−k2−1Lp(χ−1, k2−k12 + 1)Lp(χ−1, j + k

2 + 2)ζp(k1 + j + 2)

×

[(p− 1p

)(1− χ−1(p)p−(k2−k1

2 +1)) +(1− χ(p)pj+

k2 +1)

p(1− ω−1

p (p)p−j−k2−1)

]

= (p

p− 1)2ω(p)|p|−j−k2−1Lp(χ

−1, k2−k12 + 1)Lp(χ−1, j + k2 + 2)

ζp(k1 + j + 2)

×[1− χ(p)pj+

k2 − ω−1

p (p)p−j−k2−2 + χ−1(p)p−(k2−k1

2 +2)]

= ω(p)|p|−j−k2−1 Lp(χ−1, k2−k12 + 1)Lp(χ−1, j + k2 + 2)

ζp(k1 + j + 2)Lp(χ,−j − k2 )Lp(ω−1

p , j + k2 + 2)

=εp(−j − k

2 −12 , $p, ψp)εp(k1−k2+1

2 , $p, ψp)ζp(k1 + j + 2)Lp(ω−1

p , j + k2 + 2)Lp(χ−1, k2−k12 + 1)Lp(χ−1, j + k

2 + 2)Lp(χ,−j − k

2 )

=εp(−j − k

2 −12 , $p, ψp)εp(k1−k2+1

2 , $p, ψp)ζp(k1 + j + 2)Lp(ω−1

p , j + k2 + 2)Lp(f , j + k + 2)Lp(f , k2 + 1)

Lp(f,−j).

This completes the case of a special representation with minimal conductor, and thus the formula

for the bad local zeta integrals has been checked in all cases.

54

Chapter 9

Overview of Proof of Theorem4.1.1(ii)

We begin with some notation. Recall that f is a new cuspidal eigenform of weight k + 2 ≥ 2, level

N ≥ 5, and we write ω for the nebentypus of f . Choose l - 2N , and v | l a place of F . Fl denotes the

v-adic completion of F , and Ol is its ring of integers. Vl is the 2-dimensional Fl-Galois representation

attached to f , and Tl in an Ol-lattice chosen compatibly with the modular symbol δ. Recall that

δ = δ± was a Betti cohomology class on Y (N); in the notation of [Ka04], δ = δ(k + 2, k1 + 1), and

δ = δk+2,k1+1,Id. In chapter 10, we will define

z(l)1 (f , k1 + 1,−j, Id, Nl) ∈ H1(Q, Vl(t))

In the notation of that paper, σc acts on H1(Q, Vl(t)) by c−t.

9.1 Key propositions

The statement (ii) of Theorem 4.1.1 is the actually the conjunction of the following

Proposition 9.1.1. If l - N , the l-adic realization ρl,M∗(1)(ξ) is equal to

Ll(f,−j)z(l)1 (f , k1 + 1,−j, Id, Nl).

If l | N , the l-adic realization ρl,M∗(1)(ξ) is equal to

z(l)1 (f, k1 + 1,−j, Id, Nl).

Thus, according to [Ka04] §8, ρl,M∗(1)(ξ) ∈ H1(Z[ 1Nl ], Vl(t)).

Proposition 9.1.2. Assume the Main Conjecture for f and the Leopoldt-type conjecture of the

finiteness of H2(Z[ 1l ], Tl(t)). Then H1(Z[ 1l ], Tl(t)) is a rank 1 Ol-module, and the index of‘∏p|N (1−

55

appj)−1z

(l)1 (f , k1 + 1,−j, Id, Nl) · Ol in H1(Z[ 1l ], Tl(t)) is equal to #H2(Z[ 1l ], Tl(t)).

The former proposition is proved in chapter 10, and the latter in chapter 11. To deduce

Theorem 4.1.1(ii) from these propositions, recall that under the hypothesis, the canonical map

H1(Z[ 1l ], Tl(t))→ H1(Z[ 1Nl ], Tl(t)) is in fact an isomorphism, and the difference between the orders

of H2(Z[ 1l ], Tl(t)) and H2(Z[ 1Nl ], Tl(t)) is #(Ol/(

∏l|N (1− appj))Ol).

56

Chapter 10

On Constructions of Kato

In preparation to introducing K. Kato’s main conjecture for modular forms, we will review certain

constructions from [Ka04]. We then present a new proof of what is essentially a lemma of G. Kings,

introduced to study the TNC for CM elliptic curves. From this lemma, it will follow that certain of

Kato’s elements are precisely the l-adic realizations of the motivic cohomology classes constructed

in §3.2; as a consequence, Proposition 9.1.1 holds.

10.1 Kato elements

As mentioned above, the notation in this paper generally follows [HK99b], which is rather different

from the notation of [Ka04]. Furthermore, in [Ka04] there are many different cohomology classes

in many different cohomology groups; as a rule they are all labeled z, and distinguished by the

attached parameters. To maintain some degree of readability with the original, and for lack of any

truly better schema, this last convention will be continued. That being said, we begin with modular

units.

Theorem 10.1.1. ([Ka04], Proposition 1.3) Let π : E → S be an elliptic curve with identity section

E. Let c > 1 be an integer prime to 6 and invertible on S, and [c] the endomorphism of E over S

given by multiplication by c. Then there is a unique section cθ ∈ O∗(E\ ker[c]), compatible with base

change in S, so that

1. Div(cθ) = c2(e)− ker[c].

2. For any integer b prime to c, [b]∗(cθ) = cθ.

Kato applies this result to the context where S = Y = Y (M,N), E = X is its universal elliptic

curve, and c is prime to 6N . Each choice of a pair (α, β) ∈ 1MZ/Z× 1

NZ/Z determines a section of

π; let cgα,β be the pullback of cθ along this section. For c ≡ 1(mod 6MN), there is a well-defined

element

gα,β = cgα,β ⊗ (c2 − 1)−1 ∈ K1(Y )Q.

57

The motivic classes commonly called “Beilinson’s elements” will be denoted

c,dzM,N = cg 1M ,0, dg0, 1

N ∈ K2(Y (M,N)).

Lemma 10.1.1. Assume that N |N ′, M |M ′, prime(M ′) = prime(M), prime(N ′) = prime(N).

1. The trace map K1(Y (M,N ′))→ K1(Y (M,N)) takes dg0, 1N′

to dg0, 1N.

2. The trace map K2(Y (M ′, N ′))→ K2(Y (M,N)) takes c,dzM ′,N ′ to c,dzM,N .

Proof. The second statement is [Ka04], Proposition 2.3, and the first can be found in the proof given

there.

Let Hl = H ⊗Z Zl = R1π∗Zl be the relative first l-adic etale cohomology of X over Y , and

HQl= H⊗Zl

Ql. Recall that, just as the Galois representation attached to a form of weight 2, level

N , can be located in H1(YQ,Ql), similarly the representation attached to a form of weight k + 2

and level N can be located in H1(YQ,SymkHQl). Furthermore, if T denotes the relative l-adic Tate

module of X over Y , then T ∼= Hl(1). Let ξ1 and ξ2 denote the canonical sections of T/ln over

Y (Nln, Nln), and note that ξ2 still defines a section of T/ln over Y (N,Nln).

By norm-compatibility, for all M ,

cg0, 1Mlnn ∈ lim←− K1(Y (M,Mln)).

For k ≥ 0, consider the composite Chk = ChM,k (a “Chern Class map”)

lim←− K1(Y (M,Mln))ch1,1−−−→ lim←− H1(Y (M,Mln), (Z/ln)(1))

∪ξk2−−→ lim←− H1(Y (M,Mln), (SymkT/ln)(1))

Tr−−→ lim←− H1(Y (M), (SymkT/ln)(1))

∼= H1(Y (M), (SymkT )(1)).

The maps are as labelled; the first is the l-adic Chern class map, which in this context is just the

Kummer map. For c prime to Ml, define

cZM,k = Chk(cg0, 1Mlnn).

Following Kato, we do an analogous construction beginning with Beilinson’s elements. By the lemma

above, c,dzMln,Mlnn ∈ lim←− K2(Y (Mln)). We take the same partition k = k1 + k2, and j ≥ 0. (To

compare to the conventions of [Ka04], §8, his k is our k+2, his r′ is our k1 +1, and his r is our −j).

58

Recall that ζln , the primitive ln-th root of unity, is defined on Y (Mln). Consider the composite

Chk+2,k1+1,−j

lim←− K2(Y (Mln))ch2,2−−−→ lim←− H2(Y (Mln), (Z/ln)(2))

∪−→ lim←− H2(Y (Mln), (SymkT/ln)(2 + j))

∼= lim←− H2(Y (Mln), (SymkH/ln)(t))Tr−−→ lim←− H2(Y (M), (SymkH/ln)(t))

∼= H2(Y (M), (SymkHl)(t)).

Here the first labeled map is again a Chern class map, after Gillet, and the second arrow is cup

product

x 7→ x ∪ ξ⊗k11 ⊗ ξ⊗k22 ⊗ ζ⊗jln .

For c, d prime to 6lM , set

c,dz(l)M,M (k + 2, k1 + 1,−j) = Chk+2,k1+1,−j (c,dzMln,Mlnn) .

This element can be found in [Ka04], §8. We will use the same name to denote its image under

H2(Y (M),SymkHl(t))→ H1(Q,H1(Y (M)Q,SymkHl)(t)).

Choose a c, d > 1, congruent to 1 modulo Nl. For f a new cuspidal eigenform of level N and

weight k+2 ≥ 2, the classes z(l)1 (f , k1+1,−j, Id,Nl) in Proposition 9.1.1 are defined (independently

of c, d as):

• If l | N , z(l)1 (f , k1 + 1,−j, Id,Nl) is the image of

(c2 − c−k1−j)−1(d2 − d−j−k2)−1c,dz

(l)N,N (k + 2, k1 + 1,−j)

under the quotient map H1(Y (N)Q,SymkHQl)→ Vl.

• If l - N , z(l)1 (f , k1 + 1,−j, Id,Nl) is the image of

(c2 − c−k1−j)−1(d2 − d−j−k2)−1c,dz

(l)Nl,Nl(k + 2, k1 + 1,−j)

under Tr Y (Nl)Y (N) and the same quotient map.

59

10.2 Relation between Galois cohomology classes

As before, λ : X→ Y , and let λn : Xn → Y be the n-fold fiber product of X with itself over Y . The

group G = (µ2) o Σn acts on Xn, where the ith copy of µ2 acts as multiplication by −1 on the ith

copy of X, and the symmetric group Σn acts by permuting the n copies of X. Let ε : G → µ2 be

the character given by multiplication on (µ2)k, and by the sign character on Σn; the same symbol ε

will also be used to denote the corresponding projector in QG. This is the usual motivic projector

on Xn, as in [Sch90] or [Ka04].

As in [De71], the Leray spectral sequence associated to λn∗ degenerates at E2, and one may

compute Rλn∗ by the Kunneth formula. Analyzing the situation yields, for any j ≥ 0,

H1(Y (M),SymnTQl(j + 1)) ∼= Hn+1(Xn,Ql(n+ j + 1))(ε) ⊂ Hn+1(Xn,Ql(n+ j + 1)).

As before, let π1 : Xk1+j+k2 → Xk1+j be given by the first k1 + j coordinates; similarly, let π2 :

Xk1+j+k2 → Xj+k2 be given by the last j + k2 coordinates. Finally, let π : Xk1+j+k2 → Xk be given

by omitting the middle j coordinates. Let σ denote the involution of Y (M) that exchanges the two

canonical sections of X.

Proposition 10.2.1. Assume l |M . For j ≥ 0,

c,dz(l)M,M (k + 2, k1 + 1,−j) ∈ H2(Y (M),SymkHQl

(t))

is given by

ε π∗ (π∗1(σ∗cZM,k1+j) ∪ π∗2(dZM,j+k2)) .

Proof. After multiplying both sides by k!, they become elements of

H2(Y (M), (SymkHl)(k + 2 + j)), and it is enough to prove the equality for these mod ln for all n.

Recall that R1λ∗Zl ∼= T (−1).

Consider first π∗1 , which maps

H1(Y (M),Symk1+j T (1)) ⊂ H1(Y (M), T⊗(k1+j)(1))

into

H1(Y (M), Rk1+jλk1+j+k2∗ Zl(k1 + j + 1)) =

H1(Y (M),⊕

i1+···ik1+j+k2=k1+j

Ri1λ∗Zl ⊗ · · · ⊗Rik1+j+k2λ∗Zl(k1 + j + 1)).

60

It is clear that the image of π∗1 actually lies in

H1(Y (M), (R1λ∗Zl)⊗k1+j ⊗ (R0λ∗Zl)⊗k2(k1 + j + 1)).

Similarly, the image of π∗2 lies in

H1(Y (M), (R0λ∗Zl)⊗k1 ⊗ (R1λ∗Zl)⊗j+k2(j + k + 2 + 1)).

Thus π∗1(σ∗cZM,k1+j) ∪ π∗2(dZM,j+k2) is an element of

H2(Y (M), (R1λ∗Zl)⊗k1 ⊗ (R2λ∗Zl)⊗j ⊗ (R1λ∗Zl)⊗k2(k + 2j + 2))

∼= H2(Y (M), T⊗k1 ⊗ Zl(1)⊗j ⊗ T⊗k2(2)).

Mod ln, cZM,k1+j is the trace from Y (Mln,M) to Y (M) of

ch1,1(cg 1Mln ,0

) ∪ (ξ⊗(k1+j)1 ) ∈ H1(Y (Mln,M), (Symk1+j T/ln)(1)).

Similarly, dZM,j+k2 (mod ln) is the trace from Y (M,Mln) to Y (M) of

ch1,1(dg0, 1Mln

) ∪ (ξ⊗(j+k2)2 ) ∈ H1(Y (M,Mln), (Symj+k2 T/ln)(1)).

As l |M ,Y (Mln) −−−−→ Y (M,Mln)y y

Y (Mln,M) −−−−→ Y (M)

is Cartesian, and hence π∗1(σ∗cZM,k1+j)∪π∗2(dZM,j+k2) (mod ln) is the trace from Y (Mln) to Y (M)

of

ch2,2(cg 1Mln ,0

,dg0, 1Mln) ∪ (ξ⊗k11 ⊗ (ξ1 ∪ ξ2)⊗j ⊗ ξ⊗k22 ) ∈

H2(Y (Mln), (T/ln)⊗k1 ⊗ (Z/ln(1))⊗j ⊗ (T/ln)⊗k2(2)).

Applying π∗ gives the obvious map to

H2(Y (Mln), (T/ln)⊗k(j + 2)).

Finally, on this space, k!ε is the symmetrizing operator (with no denominators). In summary,

k!ε π∗ (π∗1(σ∗cZM,k1+j) ∪ π∗2(dZM,j+k2))

61

is given (mod ln) by the trace from Y (Mln) to Y (M) of

ch2,2(cg 1Mln ,0

, dg0, 1Mln) ∪

(∑τ∈Σk

τ∗(ξ⊗k11 ⊗ ξ⊗k22 )

)ζ⊗jln .

This is exactly the formula defining c,dz(l)M,M (k + 2, k1 + 1,−j).

10.3 A lemma of Kings

According to Proposition 10.2.1, in order to realize the c,dz(l)M,M (k + 2,−j, k1 + 1) as coming from

motivic cohomology classes, it is enough to do so for the cZM,κ. In fact, one has the following result

Lemma 10.3.1. (Kings) Let e2 denote the second standard section of X[M ] over Y (M), and suppose

that l |M , κ ≥ 0, and c ≥ 1 is prime to 6lM . Then cZM,κ is the l-adic realization of the Eisenstein

symbol [c2Eis(%κ(e2))− c−κEis(%κ(ce2))

].

Compare to Lemma 4.2.9 of [Ki01], using Theorem 2.2.4 of [HK99a] (their conventions on the

degree of the polylogarithm differ by 1). Using linearity and compatibility with base change and the

GL2-action, one can recover most of the statement given there. The result here is slightly weaker

in that it does not address CM endomorphisms, but is slightly stronger in specifying the sign of

both sides (necessary for potential application to equivariant questions). We will shortly provide an

alternative proof of the lemma, whose directness hopefully compensates for the weakening. Notice

that the cZM,κ have good trace-compatibility properties as the level M varies; this is one motivation

for having rescaled the horospherical map in §3.2.

Before proving the lemma, we see how it implies Proposition 9.1.1. First,

Corollary 10.3.1. When l | M and j ≥ 0, Kato’s zeta element c,dz(l)M,M (k + 2, k1 + 1,−j) is the

l-adic realization of the class in Hk+2M (Xk(M), t)(ε)

(c2 − c−k1−j( c 1 )∗)(d2 − d−k2−j( 1d )∗)ε π∗

(π∗1Eis(%k1+j(e1)) ∪ π∗2Eis(%j+k2(e2))

).

Proof. Combining Lemma 10.3.1 and Proposition 10.2.1, c,dz(l)M,M (k + 2,−j, k1 + 1) is the l-adic

realization of

επ∗(π∗1σ∗h(c2 − c−k1−j( 1

c )∗)Eis(%k1+j(e2))i∪ π∗2(d2 − d−k2−j( 1

d )∗)Eis(%j+k2(e2))) =

62

επ∗(π∗1(c2 − c−k1−j( c 1 )∗)Eis(%k1+j(e1)) ∪ π∗2(d2 − d−k2−j( 1

d )∗)Eis(%j+k2(e2))).

Since ( c 1 )∗ fixes Eis(%j+k2(e2)) and ( 1d )∗ fixes Eis(%k1+j(e1)), and since the GL2(Z/M) action

commutes with the πi and π, the corollary follows.

We will also need

Lemma 10.3.2. Suppose l - M . Let Tr be the trace from Y (Ml) to Y (M), and T ′l the dual Hecke

operator (as in [Ka04], §4.9). Then

ρl(Tr (ξMl)) = ρl((1− T ′l ( 1/l

1)∗lj + ( 1/l

1/l )∗l1+2j

)ξM ).

Proof. As both sides are sums of cup products of Eisenstein symbols, it is enough to prove the

equality with ρD in place of ρl. Write

ξ• = π∗1Eisk1+j(φ1) ∪ π∗2Eisj+k2(φ2),

so that ξM = ε π∗ξM• . As π∗ and ε commute with the GL2(Z/N) and Hecke actions, we want to

show

Tr (ρDξMl• ) =

(1− T ′l ( 1/l

1)∗lj + ( 1/l

1/l )∗l1+2j

)ρDξ

M•

Proposition 4.4 of [Ka04] proves an analogous result for Eisenstein series, using certain norm-

compatibility relations. According to [Sch98], these same relations are already satisfied in motivic

cohomology, and a fortiori in the l-adic realization, so the same argument holds (again, mind that

we have rescaled by a factor of the level).

Proof of Proposition 9.1.1. Recall the definitions of z(l)1 (f, k1 + 1,−j, Id,Nl) from above. If l | N ,

the proposition follows immediately from Cor. 10.3.1 (with M = N). If l - N , Cor. 10.3.1 gives that

at least

ρl(ξNl) = z(l)Nl,Nl(k + 2, k1 + 1,−j).

By definition, the trace from Y (Nl) to Y (N) takes z(l)Nl,Nl(k+2, k1+1,−j) to z(l)

N,N (k+2, k1+1,−j).

Since l-adic realization commutes with trace maps, by Lemma 10.3.2,

ρl

((1− T ′l ( 1/l

1)∗lj + ( 1/l

1/l )∗l1+2j

)ξN)

= z(l)N,N (k + 2, k1 + 1,−j).

63

According to [Ka04], 4.9.3, the T ′l (1/l

1)∗ commute with the GL2(Z/N)-action, so

ρl

((1− T ′l ( 1/l

1)∗lj + ( 1/l

1/l )∗l1+2j

)Tr Y (N)

Y1(N)ξN)

= Tr Y (N)Y1(N)z

(l)N,N (k + 2, k1 + 1,−j).

Now apply ⊗eTλ to both sides. As ( 1/l1) acts trivially on Y1(N), we obtain

(1− allj + ω(l)l1+2j)ρl(ξ) = z(l)1 (f, k1 + 1,−j, Id , Nl).

The coefficient on the left is Ll(f,−j)−1, and the proposition is shown.

10.4 Proof of Lemma 10.3.1

This proof was inspired by [HK99a]. If κ = 0, then the statement is essentially tautologous, so

assume κ ≥ 1. The Gysin sequence that defines the residue map in motivic cohomology defines a

similar residue map in l-adic cohomology. The residue maps commute with realization, so one has

a commutative diagram

Hκ+1M (Xκ, κ+ 1) −−−−→ Q[Isom](κ)y y

Hκ+1et (Xκ,Ql(κ+ 1)) −−−−→ Ql[Isom](κ)

which restricts to

Hκ+1M (Xκ, κ+ 1)(ε) −−−−→ Q[Isom](κ)y y

Hκ+1et (Xκ,Ql(κ+ 1))(ε)

∼=−−−−→ Ql[Isom](κ)

For the isomorphism in the last row, see the proof of [HK99a], Theorem C.2.2. Since

Hκ+1et (Xκ,Ql(κ+ 1))(ε) ∼= H1(Y (M), (Symκ TQl

)(1)),

and since H1(Y (M), (Symκ T )(1)) ∼= Mκ(Y (M)) ⊗ Zl is torsion-free, at least cZM,κ is an element

of the right space to be the l-adic realization of the given Eisenstein symbol. By the isomorphism

above, by the fact that Resκ is a left inverse to Eisκ, and by the formulae for %κ (§3.2, §5.3), is is

enough to show

Proposition 10.4.1. Let i∞ be the usual cusp at infinity with its usual orientation. Let h =

( ∗ ∗X Y ) ∈ GL2(Z). Then, assuming l |M ,

Resκ(cZκM,M )(h · i∞) =Mκ+1

κ+ 2(c2Bκ+2(

X

M)− c−κBκ+2(

cX

M)).

64

First, one needs a more explicit description of the l-adic residue map. Let Yη be the completion

of Y (M) at the standard cusp ∞. Let Yη,n be the base change of Y (Mln) to Yη

Lemma 10.4.1. Over Yη, there exists a monodromy filtration

0→ Ql(1)→ TQl→ Ql → 0.

The induced maps Symκ TQl→ Symκ−1 TQl

induce isomorphisms on residue. Furthermore, the

standard section ξ1 of T/ln over Yη,n gives a splitting of

0→ Z/ln(1)→ T/ln → Z/ln → 0.

There is a similar construction at each cusp

Proof. For the first two assertions, see [HK99a], 2.1.2, 2.1.3. The third follows from considering the

situation over C.

Let j be the inclusion of Y into Y , and let i be the inclusion of its reduced complement Cusps.

By C.2.3 and 2.1.3 of [HK99a], the residue map on H1(Y,Symκ TQl(1)) is the composite

H1(Y,Symκ TQl(1))→ H0(Cusps, i∗R1j∗Symκ TQl

) τκ

−→ H0(Isom,Ql)

where τκ is, at each cusp, the κ-fold composite of the map τ defined above. We thus have a

commutative diagram

H1(Y, SymκTQl(1)) −−−−−→ H0(Cusps, i∗R1j∗SymκTQl)τκ

−−−−−→ H0(Isom, Ql)x?? x?? x??H1(Y, SymκT (1)) −−−−−→ H0(Cusps, i∗R1j∗SymκT )

τκ

−−−−−→ H0(Isom, Zl)??y ??y ??yH1(Y, SymκT/ln(1)) −−−−−→ H0(Cusps, i∗R1j∗SymκT/ln)

τκ

−−−−−→ H0(Isom, Z/ln)

It is enough to compute the residue along the bottom row for each n. Let G be the Galois group of

Y (Mln) over Y (M). The diagram is compatible with base change to Y (Mln), so when computing

residues mod ln, we may consider our elements as G-invariant elements of H1(Y (Mln),Symκ T (1)).

These are the preliminaries necessary for the computation.

Let

0→ Z/ln(1)→ Gm → Gm → 0

65

be the Kummer sequence over Y = Y (M), Y (M,Mln), or Y (Mln), and let

δ : H0(Y,Gm)→ H1(Y,Z/ln(1))

be the first coboundary map. Recall that ξ1 and ξ2 define global sections of T/ln over Y (Mln).

Examining the definitions, one sees that

cZM,κ(mod ln) ∈ H1(Y (M),Symκ T/ln(1))

is equal toln−1∑A,B=0

δ(cg AM

Mln ,BM+1Mln

)⊗ [AMξ1 + (BM + 1)ξ2]⊗κ.

By GL2(Z)-equivariance, the residue of this at h · i∞ is equal to the residue at i∞ of

ln−1∑A,B=0

δ(h∗cg AM

Mln ,BM+1Mln

)⊗ [h∗(AMξ1 + (BM + 1)ξ2)]⊗κ.

Rearranging the summation, this equals

ln−1∑A,B=0

δ(cgAM+X

Mln ,BM+YMln

)⊗ [(AM +X)ξ1 + (BM + Y )ξ2]⊗κ.

Now base change to Yη and apply the composite Symκ T/lnτκ

−→ Z/ln. Since τ take ξ1 7→ 1 and

ξ2 7→ 0, the residue of cZM,κ at h is equal to the residue at ∞ of

ln−1∑A,B=0

δ(cgAM+X

Mln ,BM+YMln

)· (AM +X)κ ∈ H1(Yη,n,Z/ln(1))G .

Lemma 10.4.2. The residue of δ(cgα1,α2) at ∞ is M2 (c2B2(α1)−B2(cα1)).

Proof. As discussed in [HK99a], below Lemma C.3.1, given a global section f of Gm over Yη, the

residue of δ(f) is exactly the obstruction to taking a ln-th root of f at that cusp, i.e., the order of

vanishing of f at the cusp. The power series expressions for f = cgα1,α2 are given, for example, by

Kato [Ka04].

Thus

Resκ(cZM,κ)(h · i∞)(mod ln)

=M

2

ln−1∑A,B=0

(c2B2(

AM +X

Mln)−B2(

c(AM +X)Mln

))· (AM +X)κ.

To prove the proposition, hence the lemma, it is enough to show

66

Lemma 10.4.3. As n→∞,

ln

2

ln−1∑A=0

(c2B2(

AM +X

Mln)−B2(

c(AM +X)Mln

))· (AM +X)κ

converges l-adically to Mκ

κ+2 (c2Bκ+2(XM )− c−κBκ+2( cXM )).

Proof. This result is very similiar to [Wa97], Theorem 2.2, and can be proved directly as therein.

For a quicker argument, we may use results of [L76], §XIII.2,3. In the notation of loc. cit., the limit

of the left side is

c2

2M

∫IX(Y )Y κdE2,c−1 =

c2

M(κ+ 2)

∫IX(Y )dEκ+2,c−1

=c2

M(κ+ 2)

[E

(M)κ+2(

X

M)− c−κ−2E

(M)κ+2(

cX

M)]

=Mκ

κ+ 2(c2Bκ+2(

X

M)− c−κBκ+2(

cX

M)).

67

Chapter 11

On Kato’s Main Conjecture

11.1 Some Iwasawa theory

Let Gn = Gal(Q(ζln)/Q), and G∞ = lim←− Gn = Gal(Q(ζl∞)/Q). The cyclotomic character

χcyclo : G∞∼=−→ Z∗l

is defined by the action on the Tate module of l-torsion roots of unity, and for c ∈ Z∗l , let σc =

χ−1cyclo(c). The Iwasawa algebra Λ is defined, as usual, to be

Λ = lim←− Zl[Gn].

Since l 6= 2, one has

Λ ∼= Zl[(Z/l)∗][[T ]].

For the Galois module Tl chosen in §3.1, or more generally for any l-adic Galois representation,

define

HmIw(Tl) = lim←− Hm(Z[ζln ,

1p], Tl).

For any i ∈ Z, there is a canonical isomorphism HmIw(Tl) ∼= Hm

Iw(Tl(i)), and hence a canonical map

si : HmIw(Tl)→ Hm(Z[

1p], Tl(i)).

In the previous chapter, there were constructed classes

(k!)c,dz(l)Nlm,Nlm(k + 2, k1 + 1,−j) ∈ H2(Y (Nlm),SymkHl(t)).

By their definition, these are compatible under the natural trace maps. The Hochschild-Serre spectral

68

sequence yields natural arrows

β : H2(Y (Nlm),SymkHl(t))→ H1(Q,H1(Y (Nlm)Q,SymkHl)(t)),

and one may consider the (k!)c,dz(l)Nlm,Nlm(k + 2, k1 + 1,−j) as norm-compatible elements of the

latter. The norm-compatibility forces the classes to be unramified outside of l (see [Ka04], Lemma

8.5), and so

(k!)c,dz(l)Nlm,Nlm(k + 2, k1 + 1,−j)m ∈ lim←− H1(Z[

1l],H1(Y (Nlm)Q,SymkHl)(t)).

The trace maps from Y (Nlm) to Y (N)⊗Q(ζlm) induce

lim←− H1(Z[1l],H1(Y (Nlm)Q,SymkHl)(t))→ lim←− H1(Z[ζln ,

1l],H1(Y (N)Q,SymkHl)(t))

→ H1Iw(Tl)⊗Q.

We will write c,dz(l)δ (t) for the image of c,dz(l)

Nlm,Nlm(k + 2, k1 + 1,−j)m in H1Iw(Tl)⊗Q, and set

z(l)δ (t) = (c2 − ck2+2σc)−1(d2 − dk1+2σd)−1

∏p|N

(1− app−k−2σ−1p )−1

c,dz(l)δ (t).

Here δ = δkiis the same δ appearing in §3.1, and the definition of z(l)

δ agrees with Kato’s from §13.9

after a Tate twist of k+ 2. A priori, z(l)δ is defined as an element of H1

Iw(Tl)⊗Quot(Λ), but [Ka04],

§13.12 shows that in fact z(l)δ ∈ H1

Iw(Tl)⊗Q. Note also that for j ≥ 0, by definition

β(c,dz(l)N,N (k + 2, k1 + 1,−j)) = st(c,dz

(l)δ (t)) ∈ H1(Z[

1l], Vl(t)).

11.2 Statement of the Main Conjecture

By choice of the lattice Tl, δ is a generator of T±l . Let us also choose a generator γ of T∓l . Just as

we defined z(l)δ , one may also define z(l)

γ ∈ H1Iw(Tl) ⊗ Q as in [Ka04], §13.9, except with the noted

difference in convention on twisting. The exact definition of z(l)γ is not relevant, but for correctness

we need elements coming from both the plus and minus part of Tl in order to generate a submodule

that can be related to specialization at both even and odd twists.

As in [Ka04], Th. 12.5.(4), let Z(f, T ) be the Λ-submodule of H1Iw(Tl)⊗Q generated by z(l)

δ (t)

and z(l)γ (t). As σ−1 acts on z(l)

δ (t) as 1 and on z(l)γ (t) as −1, Z(f, T ) is generated over Λ by the single

element z(l)δ+γ(t) = z

(l)δ (t) + z

(l)γ (t). Kato’s l-adic main conjecture for f (l 6= 2) now reads

69

Conjecture 11.2.1 (Kato’s Main Conjecture, 12.10 of [Ka04]). Let p be a prime ideal of Λ of height

1. Then Z(f, T )p ⊂ H1Iw(Tl)p, and

lengthΛp(H2

Iw(Tl)p) = lengthΛp(H1

Iw(Tl)p/Z(f, T )p).

We remark that, for all f , the module Z(f, T ) just defined really is the module that appears in

the statement of the Main Conjecture. When f has CM, one prefers to work with elements coming

from elliptic units, but the connection between the two constructions is given in §15.16. By Rubin

[Ru91], the main conjecture for imaginary quadratic fields is known to hold in many cases; see [Ka04]

§15 for a discussion of how to relate the two results in the CM case.

11.3 Descent to twist (t)

We finish by briefly giving a descent argument, following Kato to reduce from the Main Conjecture

given above to a finite level statement.

Lemma 11.3.1. ([Ka04], Lemma 14.15) Let A be a Noetherian commutative ring, let C be the

category of f.g. A-modules M such that the support of M in Spec A is of codimension ≥ 2, and let

G(C) be the Grothendieck group of the abelian category C. Let M be a finitely generated A-module

whose support is of codimension ≥ 1, let a ∈ A, and assume that Mp = 0 for any prime ideal of

height 1 that contains a. Then M/aM and aM = ker(a : M →M) belong to C, and we have

[M/aM ]− [aM ] =∑

q

lengthAq(Mq) · [A/(q + aA)]

in G(C), where q ranges over all prime ideals of A of height 1 that do not contain a, and where [·]

denotes the class in G(C).

In the case A = Λ, C is the category of finite A-modules, and equality in G(C) implies an equality

of the orders of the groups involved. Continuing Kato’s argument, let p be the augmentation ideal

of Λ. The p is principal; let a be a generator. One has an exact sequence

0→ H1Iw(Tl(t))/aH1

Iw(Tl(t))→ H1(Z[1l], Tl(t))→ aH

2Iw(Tl(t))

and the isomorphism

H2Iw(Tl(t))/aH2

Iw(Tl(t)) ∼= H2(Z[1l], Tl(t)).

Assume for the moment that the hypotheses of Lemma 11.3.1 are met for either choice of M =

70

H2Iw(Tl(t)) or M ′ = H1

Iw(Tl(t))/Z(f, T ). The operator σc acts on Vl(t) by c−t; it follows that

z(l)γ (t) ∈ aH1

Iw(Tl(t)),

and the image of z(l)δ+γ(t) in H1

Iw(Tl(t))/aH1Iw(Tl(t)) is equal to

z =∏p|N

(1− appj)−1z(l)1 (f, k1 + 1,−j, Id , Nl).

According to the Main Conjecture, one has an equality in G(C)

[M/aM ]− [aM ] = [M ′/aM ′]− [aM ′].

Under our hypotheses, z can not be Zl-torsion in H1(Z[ 1l ], Tl(t)), so there do not exist w ∈

H1Iw(Tl(t)), b ∈ Λ−(a) such that aw = bz

(l)δ+γ(t). As H1

Iw(Tl(t)) is torsion-free, [aM ′] = 0. Forgetting

about all but the orders of the groups, one has concretely

#(H2Iw(Tl(t))/aH2

Iw(Tl(t))) ·#(aH2Iw(Tl(t)))−1 = [H1

Iw(Tl(t)) : aH1Iw(Tl(t)) : z],

and so

#(H2(Z[1l], Tl(t))) = #(H2

Iw(Tl(t))/aH2Iw(Tl(t)))

= #(aH2Iw(Tl(t)))[H1

Iw(Tl(t))/aH1Iw(Tl(t)) : z]

= [H1(Z[1l], Tl(t)) : z].

This is the desired finite level result. What does it mean that M and M ′ satisfy the hypothesis

of Lemma 11.3.1? According to the Main Conjecture, M and M ′ have the same support. The

dimension of the support of M is unchanged if M is replaced by some Tate twist, so twisting into

Kato’s range, M has support in codimension ≥ 1. It remains to require that Mp = 0 for any prime

p of height 1 that contains a. The only such prime is the augmentation ideal, for which asking that

Mp = 0 amounts to requiring that H2(Z[ 1l ], Tl(t)) be finite. In summary,

Proposition 11.3.1. Assume the Main Conjecture for f , and assume the Leopoldt-type hypothesis

that

H2(Z[1l], Tl(t))⊗Q = 0.

Then the order of H2(Z[ 1l ], Tl(t)) is equal to the index of z in H1(Z[ 1l ], Tl(t)). This is exactly the

statement of Proposition 9.1.2.

71

This completes the proof of Theorem 4.1.1. Let us conclude with a few remarks on this Leopoldt-

type hypothesis. There is an Euler system at the Iwasawa level, (whose elements were used above,

and due to Kato) which proves that the H2Iw is a torsion Λ-module. One perhaps expects that the

most natural way to prove the finiteness of H2(Z[ 1l ], Tl(t)) is to show that that elements comprising

this Euler system are nontrivial when specialized to twist t. When t is in the critical range, Kato

can relate these specializations to certain L-values, and thus prove nonvanishing. Here the interest

has been to the right of the critical strip, where the specializations are related to values of a l-adic

L-function ([Ka04], §16). And, unfortunately, the question of nonvanishing of l-adic L-functions is

still very hard.

72

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